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IN  MEMORIAM 
FLORIAN  CAJORl 


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Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

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http://www.archive.org/details/elementsofanalytOOmichrich 


''i^'—  Q^j^ 


ELEMENTS 


OF 


Analytical  Mechanics 


BY 

PETER  S.  MICHIE, 

Professor  of  Natural  and  Experimental  Philosophy  in  the 

U.   S.  Military  Academy^  West  Point,  and 

Brevet  Lieutenant-Colonel 

U.  S.  Army, 


SECOND  EDITION. 


NEW  YORK: 
JOHN     WILEY    &     SONS, 

15  AsTOR  Place. 
1887. 


Copyright,  1887, 
By  John  Wiley  &  Sons. 


N 


)^v;7 


PREFACE. 


This  volume  is  a  revised  edition  of  the  text  taught  to  the 
■cadets  of  the  U.  S.  Military  Academy  during  the  session  of 
1886-7.  Together  with  a  brief  chapter  on  Hydrodynamics,  it  is 
intended  to  comprise  a  four-months  course  of  instruction  for 
students  well  versed  in  elementary  mathematics,  including  the 
Calculus. 

The  author  has  aimed  to  present  a  clear,  concise,  yet  com- 
prehensive course,  covering  all  the  important  principles  of  Me- 
chanics which  form  the  basis  of  that  scientific  knowledge  now 
required  by  the  military  profession.  A  thorough  mastery  of  this 
volume  will  enable  the  student  to  comprehend,  upon  careful 
study,  any  of  the  more  difficult  works  upon  the  same  subject 
which  his  professional  duties  may  require  him  to  consult. 

The  table  of  contents  gives  in  consecutive  order  a  very  full 
statement  of  the  subjects  discussed.  The  arrangement  of  the 
subject-matter  and  method  of  treatment  adopted  is  in  accord 
with  the  judgment  of  several  able  scientific  officers  who  liave 
been  associated  with  the  author  in  the  instruction  of  cadets,  and 
are  the  result  of  over  sixteen  years'  experience  in  daily  contact 
with  bright  students. 

The  subject-matter  comprising  the  volume  has  been  drawn 
from  many  sources,  and  modified  to  suit  the  requirements  of  the 


IV  PREFACE. 


necessarily  limited  course.  Those  conversant  with  the  subject 
will  recognize  that  many  of  the  articles  and  illustrative  examples 
are  taken  from  Price's  Infinitesimal  Calculus,  Vols.  III.  and  IV. 
The  most  prominent  other  sources  are  Poisson's  Traite  de  Me- 
canique,  Routh's  Rigid  Dynamics,  and  Levy's  La  Statique 
Graphique. 

Lieut.  William  B.  Gordon,  Ordnance  Department,  U.  S.  Army^ 
Assistant  Professor  of  Philosophy  U.  S.  Military  Academy,  is  en- 
titled to  at  least  equal  credit  with  the  author  for  whatever  may 
be  found  worthy  of  commendation  in  the  book.  Many  of  the 
demonstrations  in  the  previous  edition  have  been  simplified  by 
him,  and  in  nearly  every  instance  where  a  question  has  arisen 
the  author  has  finally  deferred  to  Lieut.  Gordon's  better  judg- 
ment. 

The  author  is  also  under  great  obligations  to  Lieut.  Sidney 
E.  Stuart,  Ordnance  Department,  who  has  carefully  gone  over 
the  work,  and  suggested  important  changes  which,  in  most  cases, 
have  been  adopted. 


West  Point,  N.  Y.,  August,  1887. 


CONTENTS. 


MECHANICS   OF   SOLIDS. 
Introductory, 

ART.  PAGE 

1.  Mechanics  defined i 

2.  Observation  and  experiment i 

Matter. 

3.  Elements,  atom,  molecule 2 

4.  Illustration  of  the  distinction  between  elements  and  molecules 2 

5.  Mass 2 

6.  Density 2 

7.  Mass  in  terms  of  volume  and  density 2 

8.  Inertia 3 

Force. 

9.  Force  defined 3 

10.  Molecular  forces 3 

11.  Elasticity 4 

12.  Gravitation 4 

Motion. 

13.  Kinematics - 5 

14.  Motion  defined 5 

15.  Velocity;  its  measure;  parallelopipedon  of  velocity 6 

16.  Acceleration ;  resultant  acceleration 7 

17.  Angular  velocity  and  angular  acceleration 8 

Physical  Units. 

18.  The  British  system;  units  of  length,  time,  mass  and  force 8 

19.  Other  units 9 

20.  The  centimeter,  gram,  second  system 9 


VI  CONTENTS. 


Stresses  and  Motive  Forces. 

ART.  PAGE 

21.  Statics  and  kinetics lo 

22.  Measure  of  the  intensity  of  a  stress lo 

23.  Measure  of  the  intensity  of  a  motive  force 11 

24.  Impulsive  forces 12 

25.  Action  of  forces  on  free  bodies 13 

26.  Representation  of  a  force 14 

27.  Rectangular  componerits  of  a  force 15 

28.  Resultant  of  a  system  of  forces 15 

29.  The  parallelogram  of  forces 17 

30.  Application  of  the  trigonometric  functions  to  forces 17 

31.  The  resultant  of  forces  with  common  point  of  application ;  coplanar  forces  18 

32.  The  parallelopipedon  of  forces 19 

33.  The  moment  of  a  force 20 

34.  Representation  of  moments 21 

35.  Component  moments  of  a  force 21 

36.  Composition  and  resolution  of  moments 22 

37.  The  parallelopipedon  of  moments 23 

38.  Parallel  forces 24 

39.  Two  parallel  forces 25 

40.  Two  forces  whose  action-lines  are  opposite;  general  conclusions  for 

parallel  forces 26 

41.  A  couple 26 

42.  The  moment  of  a  couple 27 

Gravity. 

43.  Gravity ;  weight  of  a  body 27 

44.  Apparent  weight  of  a  body 28 

45.  Molecular  weights  considered  a  system  of  parallel  forces;  weights  pro- 

portional to  masses 28 

46.  Acceleration  due  to  gravity 29 

47.  Weight  of  the  unit  of  mass 30 

48.  The  centre  of  gravity,  the  principle  of  the  centre  of  mass;  general  for- 

mulas for  the  centre  of  gravity 31 

49.  Determination  of  the  centre  of  gravity;  by  symmetry 33 

50.  Centre  of  gravity  of  lines;  formulas  for  any  line  and  any  plane  curve; 

examples,  arcs  of  circle  and  cycloid 33 

51.  Centre  of  gravity  of  surfaces;  general  formulas 35 

52.  For   plane   surfaces;    triangle,  polygon,  circular  sector   and   segment. 

parabola  and  elliptic  quadrant 36 


CONTENTS.  Vll 


ART.  PACE 

53.  Centre  of  gravity  of  surfaces  of  revolution;  spherical  zone  and  conical 

surface 41 

54.  Centre  of  gravity  of  volumes;  general  formulas  42 

55.  Volumes  of  revolution;  paraboloid  and  spheroid 43 

56.  Centre  of  gravity  of  volumes  by  single  integration;  a  pyramid  and  cone  44 

57.  Theorems  of  Pappus .... 45 


Graphical  Statics. 

58.  Introductory 46 

59.  Reciprocal  figures  and  their  conditions 46 

60.  The  force  polygon 47 

61.  The  polar  polygon 47 

62.  The  properties  of  the  polar  polygon 49 

63.  Problems  in  graphical  statics:  (i)  general  case;  (2)  simple  Warren   truss 

with  equal  loads;  (3)  the  same  with  unequal  loads;  (4)  other  cases 52 


Work  and  Energy. 

64.  Work  done  by  a  force 60 

65.  Graphical  representation  of  work 61 

66.  Energy  defined;  kinetic  and  potential  energy 62 

67.  The  law  of  the  conservation  of  energy 64 

68.  The  principle  of  virtual  velocities 66 

69.  70.  The  fundamental  equation   of  mechanics;  reference  to  co  ordinate 

axes 67 

71.  Application  to  a  rigid  solid 69 

72.  Application  to  a  free  rigid  solid 71 

73.  Interpretation  of  equations  of  translation  and  rotation 72 

74.  Conditions  in  cases  of  constraint 73 

75.  Conditions  of  equilibrium 74 

76.  Object  of  analytical  mechanics 74 

77.  Equations  T  and  R  referred  to  centre  of  mass 74 

78.  Equations  of  translation;  conclusions 75 

79.  Illustration  of  conclusions 76 

80.  Equations  of  rotation 77 

81.  Equations  of  rotation  of  a  body  having  a  fixed  point;  a  fixed  right  line. .   78 

82.  Independence  of  the  motion  of  translation  and  rotation 79 


viii  CONTENTS. 


General  Theorem  of  Energy  applied  to  a  Free  Rigid  Solid. 
Motion  of  Translation. 

ART.  PAGB 

83.  Translation  under  incessant  forces 79 

84.  Velocity  varies  with  co-ordinates  of  centre  of  mass 80 

85.  Rotation  under  incessant  forces 81 

86.  Kinetic  energy  of  translation  and  rotation 82 

87.  Translation  under  impulsion 83 

88.  Motion  of  translation :  direct  and  inverse  problem 83. 

89.  The  direct  problem 83. 

90.  The  inverse  problem 85 

91.  Examples  of  the  direct  problem:  (i)  constant  forces 86 

92.  (2)  Motion  due  to  gravity   87 

93-   (3)  The  trajectory  in  vacuo 89 

94.  (4)  The  trajectory  in  air 94 

Motion  of  Rotation. 

95.  Moments  of  inertia 97 

96.  Radius  and  centre  of  gyration 98. 

97.  The  momental  ellipsoid 98 

98.  Principal  axes 100 

99.  Moment  of  inertia  with  reference  to  a  parallel  axis  through  the  centre 

of  mass loi 

100.  Discussion  of  the  momental  ellipsoids  of  a  body 102 

loi.   Determination  of  the  moments  of  inertia 104 

A  uniform  straight  rod;   circular  arc;  rectangular  plate;   triangle; 

elliptical  area;  ellipsoid;  rectangular  parallelopipedon. 

102.  Moments  of  inertia  tabulated 112 


Instantaneous  Axis. 

103.  Equations  of  instantaneous  axis 113 

104.  Direction  of  instantaneous  axis 114 

Rotation  of  a  Rigid  Solid  under  Impulsion. 

105.  Resultant  axis  and  plane 115 

106.  Angular  velocity  about  a  principal  axis ti6 

107.  Angular  velocity  about  the  instantaneous  and  invariable  axes. ...  116 

108.  Position  of  the  invariable  plane 117 


CONTENTS.  IX 


ART.  PAGE 

109,  The  rolling  cone 119 

no.   Discussion  of  the  rolling  cone 120 

111.  The  polhode  and  herpolhode   122 

112.  Permanent  axes  of  rotation 124 

113.  Stability  of  rotation 124 


Rotation  of  a  Rigid  Solid  under  Incessant  Forces. 

114.  Euler's  equations  of  rotation 125 

115.  Auxiliary  angles 127 

116.  Nutation  and  precessional  motion 127 

117.  The  gyroscope 128 

118.  Differential  equations  of  motion  of  the  gyroscope 130 

119.  Nutation  and  precession  of  gyroscope 132 


Impact. 

120.  Definition  of  impact;  compression  and  restitution 134 

121.  Direct  and  central  impact 135 

122.  Oblique  impact 137 

123.  Impact  against  a  fixed  obstacle 137 


Axis  of  Spontaneous  Rotation. 

124.  Equations  of  spontaneous  axis 138 

125.  Conditions  for  development  of  spontaneous  axis 140 

126.  Discussion  of  the  spontaneous  axis 14a 

127.  Axis  and  centre  of  percussion 141 

128.  Reciprocity  of  the  centres  of  percussion  and  spontaneous  rotation  for 

parallel  impacts 142 

Constrained  Motion. 

129.  Constrained  motion  defined 142 

130.  Equations  of  constraint 142 

131.  The  normal  reaction 144 

132.  The  theorem  of  energy  applied  as  in  free  motion 146 

133.  The  value  of  the  normal  reaction 147 

134.  Discussion  for  concave  and  convex  curves 149 

135.  Centrifugal  force 150 

136.  Change  in  apparent  weight  due  to  earth's  rotation 152 


CONTENTS. 


ART.  PAGE 

137.  Problems  in  constrained  motion 152 

138.  On  an  inclined  plane 153 

139.  Mechanical  property  of  circular  arcs 155 

140.  On  the  arc  of  a  cycloid 156 

141.  On  the  arc  of  a  circle 159 

Constrained  Motion  about  a  Fixed  Axis. 

142.  The  compound  pendulum 160 

143.  The  equivalent  simple  pendulum 161 

144.  Reciprocity  of  the  axes  of  oscillation  and  suspension 162 

145.  Minimum  time  of  oscillation 162 

146.  The  simple  seconds  pendulum;  Kater's  method  of  finding  its  length. .  162 

147.  Method  by  the  reversible  pendulum 165 

148.  The  value  of  the  acceleration  due  to  gravity 165 

149.  Length  of  the  equivalent  simple  pendulum 166 

150.  The  British  standard  of  length 167 

151.  Moment  of  inertia  by  the  compound  pendulum 167 

152.  The  conical  pendulum 169 

153.  The  motion  in  azimuth  of  the  conical  pendulum 170 

Equilibriujn. 

154.  Constraint  necessary  for  a  body  in  equilibrium. 171 

155.  The  three  cases  of  equilibrium;  the  first  case 171 

156.  The  second  case. 172 

157.  The  third  case I73 

158.  Case   of   a  free  body  illustrated;    examples  of  constraint — (i)  on  a 

spherical  concave  surface;  (2)  the  place  of  rest  of  a  particle  on  a  curve  173 

The  Potential. 

159.  Attractions  governed  by  the  law  of  gravitation 178 

i6o.   Component  attractions 178 

i6r.   The  potential  defined  and  explained 180 

162.  Equi-potential  surfaces 181 

163.  The  potential  expressed  in  rectangular  and  polar  co-ordinates 183 

164.  Examples   in   attractions:  a  straight   rod;  circular  arc;  circular  ring; 

circular  plate;  thin  spherical  shell;  thick  spherical  shell 184 

165.  The  theorem  of  Laplace 191 

166.  Poisson's  extension  of  Laplace's  theorem 192 

167.  Another  expression  of  the  same  theorem 193 


CONTENTS.  XI 


Motion  of  a  System  of  Bodies. 

ART.  PAGR 

168.  Translation  of  the  centre  of  mass  of  the  system 195 

169.  Translation  of  its  centre  of  mass  with  reference  to  a  fixed  origin ;  rota- 

tion of  its  centre  of  mass 196 

170.  Conservation  of  the  motion  of  the  centre  of  mass I9y 

171.  Reference  to  the  solar  system 19& 

172.  Conservation  of  moments;  invariable  axis  and  plane 199 

173.  Conservation  of  areas 2oa 

174.  Relative  acceleration 201 

175.  Differential  equations  of  the  relative  orbit 202 


Central  Forces. 

176.  Central  force  defined 203 

177.  Laws  of  central  forces 203 

178.  Differential  equation  of  the  orbit 207 

179.  Direct  and  inverse  problem  in  central  forces 208 

180.  Particular  cases  of  the  direct  problem;  central  force  attractive  and 

varying  directly  as  the  distance . , 209 

181.  Central  force  repellent 212 

182.  Central  force  attractive  and  varying  inversely  as  the  square  of  the  dis- 

tance   212 

183.  The  velocity  from  infinity  at  any  distance  R 215 

184.  The  velocity  at  any  point  of  the  orbit 216 

185.  The  time  of  description  of  any  portion  of  the  orbit;  the  elliptical  orbit ; 

the  parabolic  orbit;  the  hyperbolic  orbit 216 

186.  The  anomalies 222 

187.  Illustration  of  the  anomalies 224 


The  Solar  System. 

188.  The  solar  system  defined 225 

189.  Kepler's  laws 226 

190.  Consequences  of  Kepler's  laws 226 

191.  Law  of  universal  gravitation 228 

192.  Planetary  orbits 23a 


Xll  CONTENTS. 


THEORY  OF   MACHINES. 
Resistances. 

ART.  PAGE 

193.  Resistances  in  machines 234 

194.  Friction 234 

195.  Coefficient  of  friction 235 

196.  Problems  involving  friction 237 

(i)  Motion  on  a  plane  surface 237 

(2)  Friction  on  a  trunnion 238 

(3)  Friction  on  a  circular  pivot 240 

(4)  Friction  on  a  ring  pivot 241 

(5)  Friction  of  a  cord  on  a  cylinder 242 

197.  Stiffness  of  cordage 244 

Machines. 

198.  Machine  defined 246 

199.  Theory  of  machines 247 

200.  Use  of  fly-wheel 249 

201.  Efficiency. ...   250 

/  Simple  Machines. 

202.  Gain  and  loss  of  power;  equation  of  equilibrium 251 

203.  The  lever 253 

204.  The  common  balance 255 

205.  The  wheel  and  axle 257 

206.  The  differential  wheel  and  axle 260 

207.  The  pulleys;  the  fixed  pulley ; 260 

208.  The  movable  pulley 263 

209.  The  block  and  fall   264 

210.  Other  combinations  of  fixed  and  movable  pulleys 266 

211.  The  inclined  plane 266 

212.  The  work  done  in  moving  a  body  up  a  plane 268 

213.  The  wedge 268 

214.  The  screw. 271 

215.  The  modulus  of  the  screw 273 

216.  The  cord 275 

217.  The  differential  equations  of  the  funicular  curve 276 


CONTENTS.  xiii 


ART.  PAGE 

2i8.  The  direction  of  the  resultants  of  equilibrium  forces  acting  on  a  cord  277 

219.  The  ratio  of  the  intensities  of  these  resultants 278 

220.  Application  to  a  funicular  curve 278 

221.  The  catenary  curve 279 

222.  The  common  catenary 279 

223.  The  directrix  of  the  catenary 281 


APPENDIX  TO    MECHANICS   OF  SOLIDS. 


Table 
Table 


I.  Densities  and  specific  gravities  of  substances 283 

II.  The  metric  system 287 

Table  III.  To  convert  metric  into  U.  S.  measures,  and  conversely 288 

Table  IV.  Gravity — the  values  of  g  and  L 290 

Table    V.  Friction 291 

Table  VI.  Stiffness  of  cordage  for  white  and  tarred  rope 292 


The  Greek  Alphabet  is  here  inserted  to  aid   those  who  are  not  already 
familiar  with  it,   in  reading  the  parts  of  the  text  in  which  its  letters  occur. 


Letters. 

Names. 

A    a 

Alpha 

B  (i 

Beta 

r  A 

Gamma 

/I  d 

Delta 

E  e 

Epsilon 

Z   C 

Zeta 

H  rj 

Eta 

0  OQ 

ThSta 

I  I 

Iota 

K  K 

Kappa 

A  A 

Lambda 

Mn 

Mu 

Letters. 

Names. 

N   V 

Nu 

S    ? 

Xi 

0    o 

Omicron 

n  Tt 

Pi 

P  P 

Rho 

2  as 

Sigma 

T  r 

Tau 

r  V 

Upsiloa 

$  4> 

Phi 

Xx 

Chi 

Wt/; 

Psi 

(1  GO 

Omega, 

MECHANICS   OF  SOLIDS. 


1.  Mechanics  treats  of  the  equilibrium  and  motion  of  bodies, 
or  their  elements,  under  the  action  of  forces. 

2.  The  most  ordinary  observation  shows  that  changes  are 
constantly  taking  place  in  matter.  These  changes  are  assumed 
to  be  due  to  the  action  of  force,  and  a  complete  analysis  of  the 
various  phenomena  would  make  known  the  particular  force  or 
forces  at  work.  To  make  the  analysis  we  must  cultivate  the 
faculty  of  observation  and  acquire  skill  in  experimentation.  But 
as  no  one  can  possibly  repeat  all  the  experiments,  nor  observe 
all  the  phenomena  which  at  present  form  the  data  upon  which 
Mechanics  is  based,  we  must  accept  the  certified  facts  and  the 
conclusions  derived  therefrom,  until  we  are  sufficiently  instructed 
in  the  science  to  form  for  ourselves  a  rational  judgment  as  to 
their  truth.  In  this  we  exercise  a  proper  faith  in  the  honesty, 
accuracy  and  ability  of  those  who  have  devoted  their  lives  to 
the  study  of  the  natural  sciences.  It  is  well  to  remember  that 
the  accepted  laws  which  are  assumed  to  govern  the  changes  in 
the  state  or  condition  of  matter  can  never  be  exactly  verified  by 
experiment,  because  of  the  inaccuracy  of  the  experimenter  and 
the  imperfections  of  the  appliances  by  which  the  results  are 
measured.  But  whenever  a  stated  law  appears  to  conform  more 
nearly  to  the  observed  results  as  the  experimenter  becomes  more 
skilful  and  the  apparatus  more  perfect,  we  accept  it  as  the  gov- 
erning and  limiting  law  for  this  class  of  phenomena. 


MECHANICS   OF  SOLIDS. 


Matter. 

3.  Of  the  ultimate  nature  of  matter  we  are  ignorant;  but 
from  close  observation  of  natural  laws  it  has  been  assumed: 

(i)  That  every  material  substance  is  composed  of  one  or  more 
simple  substances  or  elements^  so  called  because  they  have  thus  far 
resisted  simplification  by  subdivision; 

(2)  Tiiat  each  of  these  simple  substances  is  composed  of  very 
minute,  but  finite  and  definite,  portions,  called  atojns ; 

(3)  That  in  any  substance,  simple  or  compound,  two  or  more 
of  these  atoms  are,  in  general,  so  united  as  to  form  the  smallest 
portion  that  can  exist  by  itself  and  remain  the  same  substance. 
This  combination  of  atoms  is  called  a  7nolecule. 

4.  To  illustrate:  hydrogen  and  oxygen  are  simple  substances, 
and  two  atoms  of  hydrogen  unite  with  one  of  oxygen  to  form  a 
molecule  of  water,  which  is  a  compound  substance.  No  quantity 
of  water  less  than  the  molecule  can  exist,  but  if  the  molecule  be 
divided  it  becomes  hydrogen  and  oxygen.  In  Mechanics  the 
molecule  is  therefore  considered  as  the  elementary  mass. 

5.  Mass  is  a  term  used  to  express  the  quantity  of  matter  in  a 
body,  and  its  numerical  value  will  depend  upon  the  quantity  of 
matter  assumed  as  the  unit  mass. 

6.  Density  is  that  property  of  a  body  by  which  the  quantity 
of  matter  in  the  unit  volume  is  determined.  It  varies  in  differ- 
ent bodies  according  to  the  nearness  and  mass  of  their  constit- 
uent molecules.  When  a  body  is  compressed  its  density  is  in- 
creased because  its  molecules  are  brought  nearer  together. 
There  are  therefore  more  of  them,  and  hence  more  mass,  in  the 
unit  volume  than  before.  The  contrary  is  the  case  when  the 
body  is  expanded.  The  density  of  a  body  is  measured  by  the 
ratio  of  the  mass  in  its  unit  volume  to  the  mass  in  the  unit  vol- 
ume of  some  standard  substance,  generally  pure  water. 

7.  The  mass  of  a  body  is  therefore  directly  proportional  to 
the  product  of  its  volume  and  its  density.     If  the  unit  mass  be 


FORCE. 


assumed  to  be  the  mass  of  a  unit  volume  of  matter  at  unit  den- 
sity, then  the  mass  of  any  homogeneous  body  will  be  given  by 
the  equation 

M=  Vd, (i) 

in  which  Mis  the  mass  in  mass  units,  F  the  volume  in  volume 
units,  and  S  the  density  of  the  body. 

8.  Ifiertia  is  defined  to  be  that  property  of  a  body  by  which 
it  continues  in  its  particular  state  of  rest  or  rectilinear  uniform 
motion  until  the  action  of  some  force  produces  a  change  in  one 
or  both  of  these  states. 


Force. 

9.  Force  is  that  which  produces,  or  tends  to  produce,  any 
change  in  the  state  of  matter  with  respect  to  rest  or  motion. 

The  intensity  of  a  force  is  its  capacity  to  produce  pressure. 
The  point  of  application  is  the  molecule  of  the  body  to  which  the 
force  may  be  considered  as  directly  applied.  The  action-line  is 
the  right  line  which  the  point  of  application  would  describe  if  it 
were  free  to  move  from  rest  under  the  action  of  the  force  alone. 

A  force  is  said  to  be  completely  given  when  its  intensity, 
action-line  and  point  of  application  are  known;  for  if  any  one  of 
these  be  varied  the  resulting  effect  of  the  force  will  in  general  be 
changed. 

10.  Molecular  Forces. — Every  molecule  of  a  body  is  assumed 
to  be  the  locus  of  a  force  which  is  sometimes  attractive  and 
sometimes  repellent  according  to  the  particular  circumstances  of 
its  development.  By  virtue  of  this  force  the  molecule  unites 
itself  with  its  adjacent  molecules,  or  tends  to  separate  itself  from 
them;  and  by  means  of  these  forces  the  mass  assumes  either  the 
solid,  liquid  or  gaseous  state,  under  particular  conditions  of 
temperature  and  external  pressure.  When  an  extraneous  force 
is  applied  to  a  body  these  molecular  forces  manifest  themselves 
and  oppose  its  action.     After  the  extraneous  force  is  withdrawn 


MECHANICS  OF  SOLIDS. 


the  molecular  forces  come  again  into  equilibrio,  and  the  mole- 
cules either  resume  their  primitive  positions  of  equilibrium  or 
assume  new  ones.  For  example,  when  a  solid  bar  is  subjected 
to  a  stress  of  elongation  the  molecules  of  consecutive  cross- 
sections  will  increase  their  distances  from  each  other,  and  during 
this  state  of  separation  the  molecular  forces  will  be  attractive; 
but  when  there  is  a  stress  of  compression  the  corresponding 
molecules  are  brought  nearer  to  each  other,  and  during  this  state 
their  forces  are  repellent.  In  either  case  the  aggregate  intensity 
of  the  molecular  forces  is  equal  to  that  of  the  applied  stress. 
Wiien  the  stress  is  withdrawn,  the  molecular  forces  are  balanced 
and  the  molecules  return  to  their  primitive  positions,  provided 
they  have  not  been  forced  beyond  their  elastic  limits. 

11.  Elasticity  is  that  property  of  a  body  by  virtue  of  which  the 
molecular  forces  restore,  or  tend  to  restore,  the  molecules  to 
their  primitive  relative  positions  when  they  have  been  moved 
from  these  positions  by  the  action  of  some  force. 

The  elasticity  is  said  to  be  perfect  when  the  body  always 
requires  the  same  force  to  keep  it  at  rest  in  the  same  volume, 
shape  and  temperature,  through  whatever  variations  of  volume^ 
shape  and  temperature  it  may  have  passed. 

Every  body  has  some  degree  of  elasticity  of  volume.  All 
fluids  possess  great  elasticity  of  volume.  If  a  body  possess  any 
degree  of  elasticity  of  shape  it  is  called  a  solid;  if  none,  a  fluid. 
While  the  elasticity  of  shape  is  very  great  for  many  solids,  it  is 
not  perfect  for  any.  The  degree  of  distortion  within  which 
elasticity  of  shape  is  sensibly  perfect  is  limited  in  every  solid; 
when  the  distortion  passes  beyond  this  limit  the  body  either 
breaks  or  receives  a  permanent  set ;  that  is,  such  a  molecular 
displacement  that  it  does  not  return  to  its  original  figure  when 
the  distorting  force  is  removed. 

12.  Gravitation. — It  is  assumed  that  any  body  in  the  universe 
attracts  any  other  body  with  an  intensity  which  varies  directly 
as  the  product  of  their  masses  and  inversely  as  the  square  of  the 
distance  which  separates  them;  also  that  this  attraction  is  mutual, 
or  that  the  intensity  of  the  attraction  of  a  body  a  for  a  body  b  is 


MOTION. 


exactly  equal  and  directly  contrary  to  that  of  b  for  a.  Let  w  and 
m'  be  the  number  of  mass  units  in  the  bodies  a  and  ^,  Fig.  i;  r 
the  distance  between  their  centres,  and  /t  the  at-  ^ 

traction  of  one  mass  unit  for  another  mass  unit  at  m  *  "J'  "  m 
a  unit's  distance  apart;  then   the  intensity  of  the  fig.  i. 

mutual  attraction  of  the  two  bodies  is  given  by 


-;'; (^) 


for,  each  of  the  mass  units  of  m  attracts  each  of  those  of  m*  with 
an  intensity  }jl  at  the  distance  unity;  and  as  this  intensity  varies 

inversely  as  the  square  of  the  distance,  with  an  intensity  —^  at  the 

•distance  r;  therefore  the  m  units  of  one  body  will  attract  the  m' 

units  of  the  other  with  an  intensity  — ^-/z,  and  conversely  the  m' 

units  will  attract  the  m  units  with  an  equal  intensity.  The 
mutual  attraction  existing  between  any  two  bodies  is  a  single 
force  whose  stress  or  motive  effect  on  each  body  can  be  deter- 
mined separately. 


Motion. 

13.  Kinematics  is  that  branch  of  pure  mathematics  in  which 
the  properties  of  motion  are  considered  without  reference  to  its 
cause.  Motion  is  the  state  of  a  body  when  it  is  changing  its 
place  with  respect  to  an  origin.  A  body  is  said  to  be  at  rest 
with  respect  to  an  origin,  or  at  relative  rest,  when  it  remains  at 
the  same  distance  and  in  the  same  direction  from  the  origin. 
Considering  the  diurnal  and  annual  motion  of  the  earth,  that  of 
the  solar  system  through  space,  and  the  proper  motion  of  the 
fixed  stars,  we  see  that  a  state  of  absolute  rest  is  unknown  for 
any  body  in  the  universe.     Rest  is  therefore  wholly  relative. 

14.  Motion  is  continuous;  for  a  body  cannot  pass  from  one 
position  to  another  without  occupying  all  intermediate  positions 


MECHANICS  OF  SOLIDS, 


in  the  path  described.  Motion  may  be  uniform  or  varied.  It  is 
uniform  when  the  moving  body  describes  equal  spaces  in  equal 
successive  portions  of  time,  no  matter  how  small  these  intervals 
of  time  may  be.  When  this  condition  is  not  fulfilled  the  motion 
is  varied. 

15.  Velocity  is  the  rate  of  motion.  Its  measure^  at  any  instant, 
is  the  distance  that  would  be  described  in  the  next  subsequent 
unit  of  time,  were  the  motion  to  continue  unchanged  during  that 
unit.  Hence  the  laws  of  uniform  motion  are  embodied  in  the 
equation 

^=^A (3) 

s  being  the  distance  described  in  the  time  /,  v  the  constant  ve- 
locity, and  /  the  units  of  time  since  the  epoch  t  •=■  o.  The  prin- 
ciples of  the  calculus  and  the  definition  of  velocity  give,  for  the 
measure  of  uniform  or  varied  velocity  at  any  instant, 

1        ds  .  . 

Velocity  is  therefore  measured  by  a  distance  along  the  direc- 
tion of  the  motion  at  the  instant  considered,  and  hence  it  may 
be  graphically  represented  by  a  right  line.  The  projections  of 
this  right  line  on  co-ordinate  axes  represent  the  component  ve- 
locities in  these  directions.  If  -^t>e  the  measure  of  the  velocity  at 

any  instant,  then  the  component  velocities  along  the  axes  will  be 

7  dx    dy    dz 

measured  by  — — ,  — ^,  —7-,  and  when  the 
dt     at    dt 


/  y^JLI 


A 


axes  are  rectangular,  Fig.  2,  their  rela- 


^^  /       j     tions  to  each  other  are  given  by. 


i 


~"~^  J  X  ds        Jdx"    .   dy"    .    dz^ 


is  _  i/dx^    \_^-^'^    \_ 
dt~^  ~dt~^'di^'^  df' 


(5> 


This  is  known  as  the  principle  of  the 
parallelopipedon  of  velocitieSy  and   may  be 
thus  stated:  If  the  three  edges  of  a  parallelopipedon  which  meet  at  one 


Fig.  2. 


MOTION. 


vertex  represent  componetit  velocities,  the  diagonal  through  this  vertex 
represents  the  resultant  velocity.  It  is  employed  in  the  resolution 
and  composition  of  velocities. 

l6.  Acceleration  is  the  rate  of  change  of  velocity.  Whether 
constant  or  variable,  it  is  measured  by  tlie  increment  of  velocity 
in  a  unit  of  time,  supposing  the  acceleration  to  remain  constant 
for  that  unit,  and  the  same  as  at  the  instant  considered.  Hence 
its  measure  is 

dv      d^s  ,,. 

"  =  ^  =  -dF (^) 

When  the  velocity  is  increasing,  the  acceleration  is  regarded  as 
positive;  and  when  decreasing,  as  negative.  Acceleration  may 
be  graphically  represented  by  a  right  line,  since  it  is  measured 
by  a  velocity,  and  the  projections  of  its  rectilinear  representa- 
tive on  co-ordinate  axes  will  represent  its  component  accelera- 

d_]x  dy  dy 
dr'  dr'di 

d's 


tions  in  these  directions.     Hence  we  have  --^,  -~,  -j-j-,  as  the 


component  accelerations  of  -j-j-  when  the  latter  is  the  accelera- 
tion along  the  right-line  path  s.  If  the  path  be  a  plane  curve, 
and  p  the  radius  of  curvature  at  any  point,  we  have  from  the 
calculus 

(d'xy  +  (dyr  +  (d'zy  -  (d's)' =  ^^. .  .  .   (7) 

Dividing  both  members  by  dt*  and  reducing,  we  have 


The    first    member   is  the  resultant  acceleration,  and  its   value 

in  the  second  member  is  compounded  of  two  accelerations  at 

d^'s 
right  angles  to  each  other;  the  one,  -7^,  in    the  direction  of  the 

path  at  the  instant  considered,  and  the  other  therefore  in  the 


8  MECHANICS   OF  SOLIDS. 

direction  of  tlie  radius  of  curvature  towards  the  centre.  When 
the  path  s  becomes  a  right  line  p  becomes  infinite,  and  the  sec- 
ond component  acceleration  zero. 

17.  Angular  velocity  is  the  rate  of  motion  about  a  centre,  and 
angular  acceleration  is  the  rate  of  change  of  the  angular  velocity. 

Representing  the  first  by  g?,  and  the  second  by  — ,  their  meas- 
ures are  given  by 

dS  ,  .  doa      d'^d  doo  ,     . 

'"  =  M'  ■    •    (9)  -^  =  ^  =  ''70'    ■    ■    ('°) 

in  which  6  is  the  angle  made  by  the  radius  vector  with  the  line 
of  reference  through  the  centre.  The  tangential  linear  velocity 
of  a  point  at  a  unit's  distance  from  the  axis  is  therefore  a  meas- 
ure of  the  angular  velocity. 

The  unit  angular  velocity  is  that  of  a  point  describing  the 
unit  angle  (57°. 29578  -j- ,  called  the  radian)  uniformly  in  a  unit 
of  time. 

Physical  Units. 

18.  The  British  System. — The  British  unit  of  length  is  the/?^/. 
It  was  first  established  as  a  standard  by  taking  it  to  be  a  certain 
fraction  of  the  length  of  the  simple  seconds  pendulum  at  Lon- 
don, but  is  now  defined  to  be  the  third  part  of  the  distance 
between  two  marks  on  the  gold  plugs  of  the  Standard  Yard  de- 
posited in  the  Exchequer  at  London. 

The  unit  of  time  is  either  the  sidereal  or  mean  solar  second^  both 
of  which  are  determined  from  the  uniform  rotation  of  the  earth 
on  its  axis.  This  unit  is  international.  Unless  otherwise  stated 
the  mean  solar  second  is  assumed  as  the  unit. 

The  British  unit  of  mass  is  a  certain  platinum  cube  called  the 
Pound,  deposited  in  the  Exchequer  at  London,  and  made  the 
standard  unit  of  mass  in  Great  Britain  by  act  of  Parliament. 
By  means  of  its  copies  the  masses  of  other  bodies  may  be  deter- 
mined. 


PHYSICAL    UNITS.  9 


The  units  of  space,  time  and  mass  are  arbitrary  units,  and 
serve  to  determine  the  other  units  of  the  system,  which  are 
•called  derived  units. 

The  Unit  of  Force. — Gauss  has  defined  the  absolute  unit  of  force 
to  be  that  force  which,  acting  on  a  given  unit  of  mass  for  a  unit 
of  time,  would  generate  in  it  a  unit  of  velocity.  By  this  defini- 
tion, the  unit  force  can  be  derived  from  any  standard  units  of 
mass,  time  and  velocity  with  equal  facility.  The  British  unit 
^f  force  is  that  force  which,  acting  on  the  Pound  mass  for  one 
second,  would  generate  in  it  a  velocity  of  one  foot  per  second. 
Since  the  Pound  is  an  invariable  mass,  this  establishes  an  in- 
variable British  standard  unit  of  force.  This  unit  is  called  the 
Pounda  I. 

19.  Other  Units. — The  unit  of  velocity  is  a  velocity  of  unit  dis- 
tance per  unit  time,  and  is,''therefore,  one  foot  per  second.  Simi- 
larly, the  unit  of  acceleration  is  an  acceleration  of  one  foot  per 
second.  The  unit  of  area  is  the  square  foot;  that  of  volume^  the 
cubic  foot;  and  that  of  density^  the  density  of  the  pound  of  mat- 
ter occupying  a  cubic  foot  of  volume. 

It  will  be  shown  later  that  the  weight  of  a  given  mass  on  the 
•earth's  surface  varies  with  the  latitude  and  the  height  above  the 
sea-level.  Therefore  weight  cannot  be  taken  as  an  invariable 
standard  of  force;  but  as  the  variations  in  the  weight  of  any  mass 
are  proportionally  small,  the  weight  of  the  British  unit  of  mass 
is  generally  taken  as  the  unit  of  force  for  ordinary  purposes. 

20.  The  C.  G.  S.  System. — The  French  or  Centimeter^  Graniy 
Second  system,  is  named  from  its  length,  mass  and  time  units. 

The  unit  of  length  in  this  system  is  the  centimeter^  derived  from 
the  meter,  which  was  formerly  supposed  to  be  equal  to  one  ten- 
millionth  of  the  quadrant  of  the  Paris  meridian  line,  but  is  now 
definitely  fixed  by  a  standard  meter  in  the  Archives  at  Paris. 

The  unit  of  mass  is  the  gram,  derived  from  the  kilogram,  which 
was  originally  defined  as  the  quantity  of  matter  in  a  liter  of 
pure  water  at  the  temperature  of  maximum  density,  but  is  now 
determined  by  existing  standards. 

The  unit  of  force  in  this  system,  called  the  Dyne,  is  that  force 


10  MECHANICS  OF  SOLIDS. 

which,  acting  on  the  gram  mass  for  one  second,  would  generate 
a  velocity  of  one  centimeter  per  second. 

Stresses  and  Motive  Forces. 

21.  Nothing  is  known  of  the  inherent  nature  of  force;  but 
the  intensities  of  forces  are  assumed  to  be  proportional  to  their 
effects  under  precisely  similar  circumstances  of  action,  and  can 
be  estimated  by  comparing  these  effects.  Forces  are  classed  as 
stresses  or  motive  forces  according  as  their  effects  are  strains  or 
changes  of  state  with  respect  to  rest  or  motion.  That  branch  of 
Mechanics  whicli  treats  of  stresses  and  their  effects  is  called 
Statics,  and  that  which  treats  of  motive  forces  is  called  Kinetics. 

22.  Measure  of  the  Intensity  of  a  Stress. — When  a  solid  bar  is 
subjected  to  a  stress  within  its  elastic  limit,  experiment  shows 
that  the  elongations  are  directly  proportional  to  the  intensity  of 
the  elongating  stress,  and  to  the  original  length  of  the  bar;  and  in- 
versely proportional  to  the  area  of  cross-section,  supposed  constant 
throughout  the  experiment,  and  to  a  coefficient  depending  on  the 
material  of  the  bar.     These  experimental  laws  are  expressed  by 

in  which  \  is  the  elongation  due  to  the  stress  whose  intensity  is 
/,  /  the  original  length  of  the  bar,  s  the  constant  area  of  cross- 
section,  and  E  a  coefficient  depending  on  the  material  of  the  bar. 
If  this  law  be  supposed  to  hold  good  for  all  longitudinal 
stresses  until  A,  =  /,  we  have 

7=^ (I2> 

for  a  bar  of  unit  area  of  cross-section.  E  is  therefore  the  inten- 
sity of  that  stress  which,  applied  in  the  direction  of  the  length, 
would  elongate  a  bar  of  unit  area  of  cross-section  to  double  its 
original  length,  or  compress  it  to  zero  length  under  the  assumed 


STUESSES  AND  MOTIVE  FORCES.  II 

law.     Making  /  =  i,  and  j  =  i,  in  Eq.  (ii),  we  have 

E={,      ........     03) 

whence  E  may  also  be  defined  to  be  the  ratio  of  the  stress  to  the 
elongation  produced  by  it  in  a  bar  of  unit  length  and  cross- 
section.  It  is  called  Young's  Modulus  or  Coefficient  of  Longitu- 
dinal Elasticity.     From  Eq.  (13)  we  have 

I^EX (14) 

Hence,  within  elastic  limits,  the  intensities  of  stresses  are  as- 
sumed to  be  directly  proportional  to  the  elongations  or  com- 
pressions which  they  would  produce  in  a  given  bar,  when  applied 
longitudinally.  This  is  the  principle  of  the  common  spring- 
balance. 

23.  Measure  of  the  Intensity  of  a  Motive  Force. — Let  F  be  the 
type  symbol  of  a  force^  and  /  that  of  its  intensity.  Experiment 
and  observation  show  conclusively  that  if  the  force  act  upon  dif- 
ferent free  masses  ;;/,  m',  m'\  etc.,  /being  constant,  then  will 

that  is,  when  a  constant  force  acts  upon  different  free  bodies^  the  accel- 
eratiotis  are  inversely  as  their  masses. 

Also  that  if  two  forces  of  different  intensity  act  upon  the 
same  free  mass,  we  will  have 

^'^^"-dT^'-dr"' ('^) 

or,  the  accelerations  are  directly  as  the  intensities  of  the  forces. 

From  the  first  of  these  principles  we  get,  for  forces  of  equal 
intensities, 

d's  ,d's'  „dV'        ^  .    , 


12  MECHANICS  OF  SOLIDS. 

-and  from  the  second,  for  forces  of  different  intensities, 

I..I^::m^:m~^, (.8) 

or 

I:I^::ni^'.m  --^ (19) 

From  Eq.  (17)  we  see  that  for  any  constant  force  the  product  of 
the  mass  and  acceleration  of  the  free  body  on  which  it  acts  is  a  cofistani 
quantity;  and,  from  Eq.  (19),  that  for  different  constant  forces  the 

intensities  are  proportional  to  such  products.     Hence  ^-jt  fulfils  all 

the  requirements  of  a  measure,  and  we  may  write 

r         ^"'  /     X 

^=^:^ (^°) 

This  evidently  measures  the  intensity  of  any  motive  force;  for, 
if  the  force  be  variable,  the  acceleration  at  any  instant  is  the 
•change  in  the  velocity  which  would  take  place  in  a  unit  of  time, 
provided  the  force  were  to  remain  constant  during  that  unit. 
Hence  the  intensity  of  a  motive  force  is  ?neasured  by  the  product  of  the 
mass  of  the  free  body  upon  which  it  acts  by  the  acceleration  due  to  the 
force.  The  product  of  a  mass  by  its  velocity  is  called  its  "  quan- 
tity of  motion,"  "  quantity  of  velocity,"  and  "  momentum  "  by 
•different  authors.     The  term  momentum  is  adopted  in  this  text. 

Since 

d'^s  dv       dimv)  ,     . 

'"iF  =  '"-dt=-dr^ ("> 

we  see  that  the  measure  of  the  intensity  of  a  motive  force  is  the 
rate  of  change  of  the  momentum  taken  with  respect  to  the  time, 

24.  Impulsive  Forces.  — K  force  which  acts  on  a  body  for  a  very 
short  time,  as  in  the  case  of  a  blow,  is  called  an  impulsive  force  or 
impulsion^  while  one  whose  action  is  continuous  is  called  an  in- 
cessant force.     If  an  impulsive  force  were  to  be  measured  as   in 


STRESSES  AND  MOTIVE  FORCES.  l^, 

the  case  of  incessant  forces,  dv  would  be  great  compared  with 

dv 
dt,  and  the  expression  —  would  generally  be  a  velocity  incon- 
veniently large.  It  would  also,  in  general,  be  impracticable  to 
measure  the  duration  of  the  action  of  such  a  force.  Hence  the 
measure  of  the  intensity  of  the  impulsion  is  assumed  to  be  the 
momentum  generated  during  the  whole  time  of  action  of  the  force,  no 
matter  how  long  or  short  this  time  may  be,  and  not  that  which 
would  be  generated  in  a  unit  of  time.  Therefore  if  /^  be  the  inten-- 
sity  of  an  impulsion,  we  have  for  its  measure 

I^  —  M—  =  Mv  =  M(v^—vX     ....     (22) 

in  which  v^  is  the  velocity  of  M  at  the  instant  the  impulsion 
began  its  action  on  the  body,  and  v^  the  velocity  when  its  action 
is  completed.  It  is  evident  that  this  is  also  the  measure  of  the 
intensity  of  an  incessant  force  which  would,  in  a  unit  of  time, 
generate  the  same  momentum  as  that  which  is  actually  produced 
by  the  impulsive  force.  There  is  therefore  no  distinction  be- 
tween incessant  and  impulsive  forces  save  that  relating  solely  to 
the  duration  of  their  action. 

25.  Action  of  Forces  upon  Free  Bodies. — A  free  rigid  solid  is  a  body 
perfectly  free  to  move  under  the  action  of  any  extraneous  torce 
whatever,  its  molecules  being  so  connected  that  no  change  of 
relative  position  is  possible  among  them.  No  such  body  occurs 
in  nature.  All  bodies  change  their  form,  either  temporarily  or 
permanently,  when  subjected  to  the  action  of  extraneous  forces  ; 
and  the  results  deduced  in  Mechanics,  under  the  supposition  that 
bodies  are  free  rigid  solids,  are  not  in  strict  accord  with  those 
observed  in  actual  masses. 

If  an  isolated  molecule  could  receive  the  action  of  an  extrane- 
ous force  without  the  counterbalancing  influences  of  other 
molecules,  it  would  immediately  begin  to  acquire  accelerated 
motion,  and  continue  to  do  so  during  the  action  of  the  force; 
after  which  it  would  move  with  uniform  motion,  until  again  sub- 


14  MECHANICS  OF  SOLIDS. 

jected  to  the  action  of  force.  This  is  the  simple  consequence  of 
the  assumed  definition  of  force.  Were  the  body  acted  on  the 
hypothetical  free  rigid  solid,  the  increments  of  velocity  of  the 
different  molecules  would  be  simultaneous. 

Let  the  body  be  a  free  solid,  but  not  rigid,  and  suppose  that 
m^  m\  etc.,  Fig.  3,  represent  a  file  of  its  molecules  along  any  direc- 
tion, their  positions  being  fixed  by  the  molec- 

^  ular  attractions  which  appertain  to  the  body. 

_  Let  the  force  F  act  on  ?n  to  move  it  towards 

r  IG.  3. 

m'.  As  soon  as  m  approaches  m'  the  mo- 
lecular forces  on  m'  will  be  unbalanced,  and  m'  will  be  moved 
towards  the  next  molecule  in  order,  seeking  a  new  position  of 
equilibrium  nearer  to  m".  In  like  manner  each  molecule  will  in 
succession  take  up  its  change  of  position  and  of  state,  until  the 
last  molecule  of  the  file,  m",  is  reached.  Some  interval  of  time 
is  therefore  required  before  the  full  effect  of  the  action  of  the 
force,  as  exhibited  in  the  motion  of  the  body  as  a  whole,  is 
manifested. 

So  long  as  the  force  continues  to  act  the  molecules  are  in  a 
state  of  strain,  being  nearer  each  other  than  they  were  before; 
and  the  distance  between  each  molecule  and  the  one  on  its  left 
is  less  than  that  between  it  and  the  one  on  its  right.  The  dif- 
ference between  the  molecular  forces  corresponding  to  these  dis- 
tances is  the  force  which  is  employed  in  giving  the  molecule  its 

acceleration,  and  its  intensity  is  measured  by  ^fi-r^- 

Thus  we  see  that  the  molecular  forces  serve  to  distribute  the 
effect  of  the  extraneous  force  throughout  the  body.  Because  of 
this  action  the  point  of  application  of  a  force  may  be  taken  anywhere 
on  its  line  of  direction  within  the  limits  of  the  body. 

26.  Representation  of  Forces. — Since  a  force  is  completely  given 
when  its  intensity,  direction  and  point  of  application  are  known 
(Art.  9),  we  may  graphically  represent  a  force  by  a  portion  of  its 
action-line  equal  in  length  to  the  number  of  units  in  the  intensi- 
ty of  the  force.  One  end  of  this  portion  is  taken  at  the  point  of 
application,  and  the  direction  of  the  action  is  indicated  by  an 


STRESSES  AND  MOTIVE  FORCES. 


15 


1=12  lbs. 
Scale  1  Inch^lO  feet  or  10  lbs. 


arrow-head,  placed  generally  at  the  other  end.     Thus  the  right 
line  in  Fig.  4  represents  a  force,  since  when  p 

the  line  is  given  the  force  is  given.  The 
scale  according  to  which  the  line  is  con- 
structed is  indicated  in  the  figure.  Fig.  4. 

It  is  sometimes  convenient,  in   order  to  avoid   confusing  a 

figure,  to  take  the  extremity  at  the  arrow-head  as  the  point  of 

application;  but  when  such  departure  from  the  general  rule  is 

made,  the  change  is  evident  from  the  construction  of  the  figure. 

27.  Rectangular  Components  of  a  Force. — Let  there   be   three 

perpendicular  cords,  AO,  BO  and 
CO,  joined  at  O,  Fig.  5,  and  let  their 
directions  be  taken  as  a  set  of  co- 
ordinate axes.  The  tensions  on  the 
cords,  caused  by  the  force  P,  will  be 
given  by  the  projections  Ox,  Oy,  Oz, 
of  the  intensity  of  jP  on  the  axes. 
This  follows  from  the  assumption 
of  the  right  line  as  the  representa- 
tive of  a  force,  and  its  truth  has 
been  conclusively  shown  by  experi- 
ment. The  forces  /",  /"'  and  /*"'  are  called  rectangular  com- 
ponents of  the  force  P. 

Let  <af,  §,  y  be  the  type-symbols  of  the  angles  made  by  the 
action-line  of  a  force  with  the  co-ordinate  axes  x^y^  z  respective- 
ly, and  let  X'y  y,  Z'  represent  the  component  intensities  of  /  in 
the  directions  of  these  axes.     Then  we  have 


Fig.  s. 


X'  =  /cos  a; 
Y'  =  /cos/?; 
Z '  =  /  cos  y. 


(23) 


28.  Resultant  of  a  System  of  Forces.  The  resultant  of  a  system 
of  forces  is  a  single  force  which  will  produce  the  same  effect  as 
all  the  forces  of  the  system  acting  together.  The  forces  of  the 
system  are  called  components  of  the  resultant. 


1 6  MECHANICS  OF  SOLIDS. 

It  is  obvious  that  while  a  system  of  forces  may  have  but  a 
single  resultant,  the  resultant  may  have  any  number  of  systems 
of  components. 

Let  H  be  the  intensity  of  the  resultant,  and  a,  b,  c  the  angles 
which  its  action-line  makes  with  the  co-ordinate  axes  respec- 
tively. Since  the  effect  of  the  resultant  is  the  same  as  that  of 
the  system  of  forces,  its  component  in  any  direction  must  be  equal  ta 
the  sum  of  the  components  of  the  forces  of  the  system  in  that  direction^ 
and  we  have 

7?  cos  ^  =  /cos  or  +  /'  cos  «'  +  etc.  =  '2X*  =  X;  ) 

i?  cos  ^  =  /  cos  /?  +  /'  cos  /?'  +  etc.  =  -^Y'  =  F;  V    (24) 

R  cos  c  =  /cos  ;/  4-  /'  cos  y*  -\-  etc.  =  ^Z '  =  Z.  ) 

Squaring  and  adding,  we  have 

R^  (cos"  a  +  cos''  b  +  cos'  r)  =  X"  +  F''  +  Z^  .     .     (25) 
or 

R  =  VX'  +  y'  +  Z" (26) 

From  Eqs.  (24)  we  have 

cos  «  =  -^;      cos  ^  =  ■^;      cos  r  =  — .  .     .     .     (27) 

The  equations  of  a  right  line  making  the  angles  a^  b^  c  with 
the  axes  are 

cos  a  cos  b          cos  c' 

If  the  co-ordinates  of  the  point  of  application  of  the  resultant 
be  x\y\  z\  the  equations  of  its  action-line  become 


or 


x-x'        y-y'       z-^ 


SmESSES  AND  MOTIVE  FORCES. 


17 


29.  The  Parallelogram  of  Forces. — Let  the  system  be  composed 
of  two  forces  with  a  common  point  of  application.  Take  this 
point  as  the  origin,  Fig.  6,  and  let  the 
forces  F'  and  F"  lie  in  the  plane  XY. 
The  rectangular  components  of  F'  are 
ox*  and  oy\  and  those  of  F'*  are  ox** 
and  oy"y  neither  having  any  component 
perpendicular  to  their  plane.  The 
rectangular  components  of  the  resul- 
tant are 


ox'  -\-  ox"  =  ox; . 


(31) 
(32) 


and  the  resultant  of  F*  and  F**  is  the  resultant  of  ox  and  oy^ 
which  is  F- .  But  we  see  from  the  construction  that  F  is  the 
diagonal  of  a  parallelogram  constructed  on  F*  and  F**^  and  that 
it  passes  through  their  common  point. 

We  therefore  conclude  that  if  two  forces  have  a  common  point  of 
application^  and  a  parallelogram  be  constructed  on  their  linear  repre- 
sentatii^s,  -their  resultant  is  completely  represented  by  that  diagonal  of 
the  parallelogram  which  passes  through  their  point  of  application. 

Let  F  be  taken  to  coincide  with  the 
axis  of  F,  as  in  Fig.  7.  The  components 
of  F*  and  F**  in  the  direction  of  X  are 
equal  to  each  other  in  intensity,  and  as 
they  act  in  opposite  directions  they  coun- 
teract each  other.  The  sum  of  the  inten- 
sities of  the  components  in  the  direction 
of  Y  is  represented  by  the  length  of  the 
diagonal  of  the  parallelogram  constructed 
on  F*  and  F**,  and  we  have  the  resultant 
of  the  system  represented  by  that  diagonal, 
as  before. 

30.  Since  any  side  of  a  triangle  is  the  diagonal  of  a  parallelo- 
gram constructed  on  the  other  two  sides,  we  conclude  that  if 
two  sides  of  a  triangle  represent  the  intensities  and  directions  of 


Fig.  7. 


t^v 


t^ 


\L^ 


rrtO^ 


i8 


MECHANICS   OF  SOLIDS. 


two  component  forces,  the  other  side  will  represent  the  intensity 
and  direction  of  their  resultant.  We  assume,  therefore,  that  all 
the  trigonometric  relations  existing  between  the  angles  and  sides 
of  a  triangle  are  true  of  the  directions  and  intensities  of  a  resul- 
tant and  its  pair  of  components. 
These  relations  are 


i?»  =  /'»  +  /"«  +  2/'  /"  cos  (0'  +  0"); 
R  sin  0"   ^ 


\^ 


sin  d    ' 
R  sin  0' 


sin  d    ' 
/'  :/"  ::sin  0"  :  sin  0'; 


sin  \ 


sin 


0'  =-/( 


ios  i  <y 


=  /( 


'(5- 

-I')(,S- 

-H) 

^/' 

'(•s- 

-J")(S- 

-£) 

RI" 

\s- 

-I'){S- 

I") 

/'/' 


(33) 
(34) 

(36) 


in  which  0',  0"  and  (5^  are  the  angles  which  i"  makes  with  -/?, 
/>"  with  -ff,  and  /*'  with  /*",  respectively,  and  »S  =  — "^ — ~ — . 


31.  Any  number  of  forces  having  a  common  point  of  applica- 
tion, or  any  number  of  forces  lying  in  the  same  plane  but  having 
different  points  of  application,  may  be  combined  by  the  paral- 
lelogram of  forces.  Thus,  in  the  first  case,  find  the  resultant  of 
any  two,  then  combine  this  resultant  with  one  of  the  other  forces, 
and  so  on  until  all  have  been  combined.  The  last  resultant  is 
the  resultant  of  the  system. 

In  the  second  case,  prolong  the  action-lines  of  two  of  the 


STRESSES  AND  MOTIVE  FORCES. 


19 


forces  until  they  meet.     Take  their  intersection  as  a  common 
point  of  application,  (Art.    25),   and   then   proceed   as  above. 


Fig.  8. 

Fig.  8  is  an  illustration  of  such  a  combination,  5  being  the  re- 
sultant of  I,  2,  3  and  4. 

32.  The  Parallelopipedon  of  Forces. — The  principle  of  the  paral- 
lelogram of  forces  is  readily  extended  to  that  of  \}[i^ parallelopipe- 
don of  forces.     Thus,  in  Fig.  9,  P*"  is  the  resultant  of  P*  and  P"^ 


Fig.  g. 

and  P  of  P^"  and  /*'".  P  is  therefore  the  resultant  of  P',  P" 
and  P"\  This  principle  is  thus  stated:  If  three  forces  have  a 
common  point  of  application,  and  a  parallelopipedon  be  constructed  on 
their  linear  representatives,  their  resultant  is  completely  represented  by 
that  diagonal  of  the  parallelopipedon  which  passes  through  their  point 
of  application. 


20  MECHANICS  OF  SOLIDS. 

•33.  The  Moment  of  a  Force. — If  a  body  be  free  to  rotate  about 
a  fixed  point  it  is  evident  that  any  force  whose  action-line  does 
not  pass  through  the  point  will  produce 
such  rotation.  Thus  if  ^,  Fig.  10,  be  a 
fixed  point,  the  force  P  tends  to  ro- 
tate the  body  about  0.  This  tendency  is 
directly  proportional  to  the  intensity  of  | 
Fig.  10.  the  force  and  to  the  perpendicular  dis- 1 

tance  from  the  action-line  of  the  force  to  the  point,  and  hence 
to  their  product,  Ip.  The  product  Ip  is  therefore  the  measure 
of  the  capacity  of  the  force  to  produce  rotation  about  0,  and  is 
called  the  moment  of  P  with  respect  to  o.  Ip  is  also  the  moment 
of  P  with  respect  to  an  axis  through  0  and  perpendicular  to  the 
plane  of  0  and  the  action-line  of  the  force.  This  axis  is  called 
the  moment  axis  of  the  force  with  respect  to  0.  The  point  0  is 
called  the  centre  of  the  moment,  and/  the  lever-arm  of  the  force 
or  of  the  moment. 

To  find  the  moment  of  P  with  respect  to  any  other  axis 
through  Oy  we  make  use  of  the  principle  that  the  moment  of  a 
force  with  respect  to  any  axis  is  equal  to  the  sum  of  the  moments 
of  its  components.  The  moment  of  any  component  in  the  plane 
of  the  axis  is  zero.  Hence  if  we  resolve  the  force  into  two  compo- 
nents, one  of  which  is  in  a  plane  containing  the  axis  and  the  other 
perpendicular  to  this  plane,  the  moment  of  the  force  is  equal  to 
the  moment  of  the  perpendicular  component.  We  therefore 
have  the  following  rule  for  obtaining  the  moment  of  a  force 
with  respect  to  an  axis:  Resolve  the  force  into  two  components^  one  of 
which  shall  be  perpendicular  to  the  axis  and  the  other  in  a  plane  con- 
taining the  axis;  multiply  the  intensity  of  the  perpendicular  component 
by  the  perpendicular  distance  between  its  action-line  and  the  axis.  Any 
axis  which  is  oblique  to  the  action-line  of  the  force  is  called  a 
component  axis. 

It  is  generally  most  convenient  to  multiply  the  distance  from 
the  point  of  application  to  the  axis,  by  the  intensity  of  that  com- 
ponent of  the  force  which  is  normal  to  the  plane  containing  the 
point  of  application  and  the  axis. 


STRESSES  AND   MOTIVE  FORCES. 


21 


34,  Representation  of  Moments. — We  may  write 


//  =  /'  X 


/'. 


(37) 


Fig, 


That  is,  the  moment  measures  the  intensity  of  a  force  which  would, 
with  a  lever-arm  of  unity,  have  the  same  effect  as  the  given  force 
to  produce  rotation  about  the  axis.  Therefore  moments  and 
forces  may  be  measured  by  the  same  unit,  and  a  definite  portion 
of  a  right  line  may  be  taken  to  represent  a  moment.  To  indicate 
the  centre  of  the  moment  and  the  axis,  the  moment  representa- 
tive is  laid  off  from  the  centre  of  the  moment  on  the  axis.  To 
show  the  direction  of  motion  about  the  axis  the  line  is 
terminated  by  an  arrow-head,  and  is  laid  off  in  such  a 
direction  from  the  centre  of  the  moment  that  the  motion 
shall  appear  right-handed  as  one  looks  along  the  axis 
from  the  arrow-head  toward  the  centre  of  the  moment. 
Thus,  in  Fig.  11,  the  right  line  indicates  the  intensity  of 
the  moment,  the  axis,  the  centre  of  the  moment,  and  the  direction  of 
rotation. 

35.  The  Moments  of  a  Force  with  respect  to  Co-ordinate  Axes. — 
To  find  expressions  for  the  moments  of  a  force  with  respect  to 
any  set  of  rectangular  co- 
ordinate axes,  let  P,  Fig.  12, 
represent  the  force,  and  x*,y\ 
z'  the  co-ordinates  of  its  point 
of  application.  /  cos  y  has 
no  moment  with  respect  to 
the  axis  of  z,  and  the  moment 
of  P  with  respect  to  this  axis 
is  evidently  equal  to  the  sum 
of  the  moments  of  /  cos  a  and 
/  cos  p.  The  lever-arm  of 
/  cos  a  is  j',  and  hence  its  mo- 
ment is  /cos  ay'.  The  lever- 
arm  of  /  cos  /?  is  X* ,  and  its 
moment  is  I  cos  fix' .  These  two  moments  tend  to  produce 
rotation  in  opposite  directions.    Calling  moments  positive  when 


Fig. 


22 


MECHANICS  OF  SOLIDS. 


their  representatives  are  laid  off  in  positive  directions  along  the 
axes,  we  have  for  the  moment  of  F  with  respect  to  z 


I  {x'  cos  P  —  y'  cos  a). 


(38) 


Similarly,  for  the  moments  with  respect  to  the  axes  of  y  and 


X.  we  have 


/  (2'  cos  a  —  X*  cos  y)\ 
I  (y  cos  y  —  z'  cos  /?) 


:1 


{Z<)) 


For  a  system  of  forces  we  have 


R  (x  cos  b  —  y  Qo^  a)  =  ^2  I  (x'  cos  P  —  y*  cos  a)  —  L 
R  (z  cos  a  —  X  cos  c)  —  "2  I  {z'  cos  or  —  ;c'  cos  y)  =  M 
R  {y  cos  ^  —  -s  cos  b)  =  2  /  (y^  cos  y  —  z'  cos  /J)  =  iV^; 


(40) 


in  which  R  represents  the  intensity  of  the  resultant,  x,y  and  z 
the  co-ordinates  of  its  point  of  application,  and  ^,  b  and  c  the 
angles  which  its  action-line  makes  with  the  axes. 

36.   Composition  and  Resolution  of  Moments. — Let  oM,  Fig.  13, 

^     represent  a  moment,  and  let  it  be  required  to 

find  its  component  with   respect  to  any  axis 

through  the  centre  of  moments,  as  oM* .     We 

may  assume  that  the   moment  oM  is  due  to 

1^'    a  force  applied  at  0"  and    perpendicular   to 

^o''  the  plane  of  the  figure.     This  force  will  have 

■  ^^'  the    lever-arm  o'o*'  with    respect   to   the  axis 

oM\  and  its  moment  with  respect  to  that  axis  is  therefore,  00'^ 

being  represented  by/, 


/  (/  cos  a)  =  (Ip)  cos  a  =  oM  cos  or. 


(41) 


But  oM  cos  a  is  the  projection  of  oM  on  the  axis  oM\  Hence 


STRESSES  AND  MOTIVE  FORCES.  23 

we  see  that  the  component  of  a  moment,  with  respect  to  any  axis 
through  the  centre  of  the  moment,  is  represented  by  the  projec- 
tion of  the  moment  on  that  axis.  From  the  assumption  of  the 
rectilinear  representative  of  a  moment  we  see  that  the  princi- 
ples of  the  parallelogram  and  parallelopipedon  of  moments 
follow,  as  in  the  case  of  forces.  Figs.  6,  7  and  9  may  therefore 
be  taken  to  illustrate  the  composition  of  moments,  P,  F\  P'^  and 
P"'  representing  moments  with  a  common  centre  at  0. 

37.  Assuming  a  common  centre  of  moments  for  a  system  of 
forces,  let  Rp  represent  the  resultant  moment,  R  being  the  in- 
tensity of  the  resultant  of  the  forces,  and  /  its  lever-arm;  and 
let  /,  /«,  n  be  the  angles  which  the  moment  axis  of  the  resultant, 
called  the  resultant  axis,  makes  with  the  co-ordinate  axes  2:,^,  x 
through  the  centre. 

Then  the  principle  of  iht  parallelopipedon  of  moments  gives 


Rp^VV^M'-]-N' (42) 


That  is,  the  resultant  moment  of  a  system  of  forces  with  respect  to 
any  assumed  centre  of  moments  is  equal  to  the  square  root  of  the  sum 
of  the  squares  of  the  sums  of  the  moments  of  the  forces  of  the  system 
with  respect  to  any  set  of  rectangular  axes  through  the  assumed  centre, 
Rp  is  constant,  but  Z,  M  and  N  will,  in  general,  change  with 
the  axes. 

For  the  rectangular  components  of  Rp  we  have 


n  =  N^,\ 
m  =  M;[ 
I  =Z.   ) 


Rp  cos  n  =  JV; 

Rp  cos  m  =  M;\' .     (43) 

i^cos 


That  is,  the  algebraic  sum  of  the  moments  of  the  forces  of  the  system 
with  respect  to  any  axis  through  the  assumed  centre  is  equal  to  the  pro- 
jection of  the  resultant  moment  on  that  axis. 


24 


MECHANICS  OF  SOLIDS 


The  position  of  the  resultant  axis  is  given  by 


M 
Rf 


cos  n 


cos  m 


(44) 


38.  Parallel  Forces. — Assuming  a  system  of  parallel  forces,  we 
have 

a  =  a'  =  a"  =  etc.; 
b  =  ft'  =  /3"  =  etc.; 
c  ^=  y'  =  y"  z=z  etc.; 

and  casting  out  the  common  factor  from  Eq.  (24),  we  get 

^  =  21, (45) 

the  intensities  of  those  forces  whose  direction  cosines  are  posi- 
tive being  multiplied  by  +  i,  and  those  whose  direction  co- 
sines are  negative  by  —  i.  The  intensity  of  the  resultant  of  a 
system  of  parallel  forces  is  therefore  equal  to  the  sum  of  those 
intensities  which  act  in  one  direction  diminished  by  the  sum  of 
those  which  act  in  the  opposite  direction.  The  direction  of  the 
resultant  is  that  of  the  intensities  whose  sum  is  the  greater. 

The  point  of  application  of  the  resultant  is  found  from  the 
principle  that  the  moment  of  the  resultant  is  equal  to  the  sum  of 
the  moments  of  its  components.  Resuming  Eqs.  (40)  we  have, 
for  parallel  forces, 


(Rx  —  2Ix')  cos  b 
\rz  —  2/z')  cos  a 
{Ry  -  2//)  cos  c  =  {Rz  -  2Iz')  cos  b. 


{Ry  -  2/y)  cos  a;) 

(Rx  -  2Ix')  cos  c\\    .     .     (46) 


Since  these  equations  must  be  satisfied  for  all  possible  values 


STRESSES  AND  MOTIVE  FORCES. 


25 


of  a,  b  and  c^  the  principle  of  indeterminate  coefficients  applies, 
and  we  have 


Rx-  :2Ix'  =  o; 
jRy  -  2/y  =  o; 
Rz  -  2Iz'  =  o. 


(47) 


Hence 


2/x' 


2  = 


R 


(48) 


These  values  are  independent  of  the  angles  a,  d,  c,  and  will  be 
the  same  no  matter  what  be  the  direction  of  the  parallel  forces. 
The  point  defined  by  Eqs.  (48)  is  therefore  called  the  centre  of  the 
system.  The  position  of  the  centre  depends  on  the  intensities  of 
the  components  and  resultant,  and  upon  the  points  of  application 
of  the  components. 

If  the  points  of  application  of  the  component  forces  be  in  the 
same  plane,  as  xy,  then  2:  =  o,  and  the  centre  of  the  system  is  in 
that  plane.  If  the  points  of  application  be  on  the  same  right 
line,  as  the  axis  of  x,  then  y  ■=  z  =  o,  and  the  centre  is  on  the 
right  line  also. 

39.  Assume  a  system  of  two  parallel  forces.  If  we  take  the 
moments  of  the  components  with  respect  to  a  point  on  their  ^x 
resultant,  the  sum  of  these  moments  must  be 
zero  since  the  moment  of  the  resultant  with 
respect  to  this  point  is  zero.  If  the  two  forces 
act  in  the  same  direction,  the  sum  of  their 
moments  can  be  zero  only  with  respect  to  some 
point  between  their  action-lines,  and  we  have, 
Fig.  14,  Fig.  14. 


26  MECHANICS  OF  SOLIDS. 

rp'  -  ry^  =  o; (49> 

r:I-y.p-:f; (50) 


40.  When  the  forces  act  in  opposite  directions  the  sum  of 
their  moments  can  be  zero  only  with  respect  to  some  point  out- 

Zp"  side  the  components  and  nearer  the  greater^ 

and  we  have,  Fig.  15, 

?4fe,/  /y-/"/'  =  o;.    .    .    .    (52) 


We  conclude,  therefore, 

(i)  That  the  intensities  of  the  components  are  inversely  as 
the  distances  of  their  action-lines  from  that  of  the  resultant. 

(2)  That  the  intensity  of  either  component  is  to  that  of  the 
resultant  as  the  distance  of  the  action-line  of  the  other  compo- 
nent from  that  of  the  resultant  is  to  the  distance  between  the 
action-lines  of  the  components. 

(3)  That  the  intensities  of  any  two  of  the  three  forces  are  in- 
versely as  the  distances  of  their  action-lines  from  that  of  the 
other. 

(4)  The  resultant  lies  nearer  the  component  of  greater  in- 
tensity. 

(5)  If  three  parallel  forces  be  in  equilibrio,  the  intensities  of 
any  two  are  inversely  as  the  distances  of  their  action-lines  from 
that  of  the  other. 

(6)  The  force  of  greatest  intensity  lies  between  the  other  two. 
41.  A  Couple. — A  couple  is  a  pair  of  equal  parallel  forces  act- 
ing in  opposite  directions  but  not  immediately  opposed.     The 


GRAVITY. 


27 


perpendicular  distance  between  the  action-lines  of  the  forces  is 
called  the  arm  of  the  couple.     From  Eq.  (54)  we  have 


r  = 


_r{r-f) 


r  -r 


(55> 


Let  /"  approach  /'  in  value,/"  — /'  remaining  constant.  Then 
as  /'  —  /"  diminishes,/"  increases,  and  when  the  two  forces  be- 
come a  couple/"  becomes  infinite,  and  R  becomes  zero.  Hence 
the  resultant  of  a  couple  is  a  force  of  zero  intensity  at  infinity; 
that  is,  no  single  force  can  replace  a  couple. 

42.  The  moment  of  a  couple  with  respect  to  any  centre  in 
the  plane  of  the  forces  is  equal  to  the  product  of  the  arm  of  the 
couple  by  the  intensity  of  one  of  the 
forces.  To  show  this  let  the  couple  be 
as  indicated  in  Fig.  16,  and  assume  the 
three  points  ^,  o\  0"  as  centres.  Then 
we  have 

rp>  +i"p''=l'(p'  +/")  =  /'/; 

I'f    _  I"pvi  =  /'(pv   _^H)     ^  f,p_  )  j,__^    _^ 


(56) 


A  couple  is  represented   by  its   moment,  and  hence  couples, 
may  be  combined  in  the  same  manner  as  moments. 


Gravity. 

43.  The  attraction  of  the  earth  for  any  part  of  its  own  mass 
is  called  Gravity;  it  is  a  special  case  of  universal  gravitation. 

The  weight  of  a  body  is  the  resultant  of  all  the  forces  of 
gravity  acting  on  its  molecules,  and  it  will  be  known  when  its 
intensity,  action-line,  and  point  of  application  are  known.  The 
weight  may  act  either  as  a  stress  or  as  a  motive  force.  In  the 
former  case   each   molecule   presses  on  the  one  below,  so  that. 


28  MECHANICS  OF  SOLIDS. 

each  horizontal  stratum  of  molecules  of  a  body  at  rest  is  sub- 
jected to  the  stress  arising  from  the  weight  of  all  above  it,  and 
the  body  is  subjected  to  a  compressive  strain.  When  the  body 
is  free  to  move,  the  weight  of  each  molecule  causes  its  own 
acceleration,  and  none  of  the  weight  acts  as  a  stress. 

44.  From  geodetic  and  astronomical  observations  the  earth 
is  found  to  be  an  oblate  spheroid  whose  polar  semi-axis  is  ap- 
proximately 3949.55  ±  miles,  and  equatorial  radius  3962.72  ± 
miles.  Were  there  no  rotation  of  the  earth  about  its  axis,  the 
apparent  weight  of  a  molecule  would  be  that  due  to  the  earth's 
attraction,  and  it  would  be  directed  to  the  earth's  centre.  But 
owing  to  this  rotation  the  molecule  is  constantly  carried  along 
the  circumference  of  its  circle  of  latitude,  and  this  can  only  be 

the  case  when  a  force  normal  to  the  tangent  and 

directed  towards  the  centre  of  the   circle  acts 

fnv^ 
upon  it  with  an  intensity  —  (Arts.  16  and  23). 

Let  w,  Fig.    17,  be  the   molecular  mass,  ml  the 

plane  of  its  circle  of  latitude,  mg  its  weight,  and 

tnv^ 
Fig.  17.  ^-^  ~  —  ^^  force  which    is   just   sufficient   to 

cause  it  to  continue  on  the  circumference  of  the  circle  of  lati- 
tude. We  see  that  nip  is  that  component  of  the  weight  of  m 
employed  in  deflecting  it  from  its  rectilinear  path,  and  the  other 
component  alone  causes  pressure  on  what  supports  m.  This 
component,  7ng\  is  called  the  apparent  weight,  and  is  not  directed 
towards  the  centre  of  the  earth  except  when  in  is  at  one  of  the 

poles  or  on  the  equator.     The  maximum  value  of  —  due  to  the 

angular  velocity  of  the  earth  is  only  about  ^-^  that  of  mg  ;  we 
may  therefore,  for  the  present,  neglect  its  consideration  and 
assume  that  the  direction  of  the  weight  is  towards  the  earth's 
centre. 

45.  Again,  since  the  longest  dimension  of  any  body  whose 
weight  is  to  be  found  is  insignificant  compared  with  the  earth's 
radius,  the  action-lines  of  the  molecular  weights  of  any  body 


GRA  VI TY.  29 


may  be  taken  parallel  to  each  other,  and  the  system  of  molecu- 
lar weights,  therefore,  to  be  a  system  of  parallel  forces. 
From  Eq.  (2)  we  may  write 

_       (M—  m)m  .     . 

G=- -^T-^l^^ (57) 

and  considering  M  to  be  the  mass  of  the  earth  and  m  that  of  the 
body  whose  weight  is  to  be  found,  we  see  that,  owing  to  the 
great  relative  mass  of  the  earth  as  compared  with  ///,  in  all  prac- 
tical applications  M  —  m  may  be  taken  as  a  constant,  and  the  in- 
tensity of  the  weight  will  therefore  vary  directly  with  w,  and 
inversely  as  r^  Since  the  attraction  is  mutual  the  mass  m  at- 
tracts the  earth  with  the  same  intensity  that  the  earth  attracts  m^ 

The  variation  in  the  weights  of  unit  masses  in  any  body  at 
the  same  locality,  due  to  their  increased  distances  from  the 
centre  of  the  earth,  can  be  neglected.  For,  assuming  the  differ- 
ence of  distance  to  be  one  mile,  the  radius  of  the  earth  being 
taken  as  4000  miles,  the  weights  of  the  same  body  will  be  as 
(4001)''  :  (4000)"  or  I  :  1.0005  »  ^^^^  is,  there  is  an  increase  or 
diminution  of  its  weight  by  ^^jVtt  P^^^  ^"^  ^^  ^  decrease  or  in- 
crease of  a  mile  in  distance  from  the  centre  of  the  earth.  There- 
fore the  weights  of  the  different  unit  masses  of  all  bodies  whose 
weights  are  to  be  found  may  be  taken  to  be  sensibly  equal  to 
each  other. 

46.  The  effect  of  gravity  on  a  free  body  is  to  cause  it  to  fall 
toward  the  earth  with  a  constantly  increasing  velocity.  Repre- 
senting the  weight  of  the  body  by  a/,  and  its  mass  by  w,  we 
have  (Eq.  20) 

d^s              d'^s       w  .  _. 

""  =  "'!?'     "'     W=m (58) 

From  this  we  see  that  since  the  ratio  of  the  weights  of  all 
bodies  to  their  masses  is  a  constant  at  the  same  place  on  the 
earth's  surface,  their  accelerations  caused  by  the  earth's  attrac- 
tion is  also  a  constant  at  the  same  place. 


30  MECHANICS  OF  SOLIDS. 

This  acceleration  is  called  the  acceleration  due  to  gravity^  and 
is  represented  in  the  text  by^.  Its  value  has  been  determined 
by  much  careful  experiment,  and  is  given  by  the  equation. 

^  =  32.173  —  0.0821  cos  2\  —  0.000003^,  .    .     .     (59) 

in  feet  per  second,  or 

g  =  980.6056  —  2.5028  cos  2X  —  0.000003^,    .     .     (60) 

In  centimeters  per  second,  in  which  X  is  the  latitude  of  the  place 
^nd  ^  the  height  above  the  sea-level. 

47.  The  unit  of  mass  depends  on  the  unit  of  intensity  and 
is,  by  definition,  that  quantity  of  matter  which  acquires  a  unit 
■of  velocity  when  acted  upon  by  a  force  of  unit  intensity  for  a 
unit  of  time.  The  weight  of  the  pound  mass  being  the  unit 
•of  intensity  in  ordinary  use,  it  is  necessary  to  determine  the 
weight  of  the  corresponding  unit  of  mass.  When  one  pound 
intensity  acts  on  one  pound  mass,  which  is  the  case  when  a  body 
falls  freely  in  vacuo,  the  acceleration  is  g  feet  per  second;  when 
one  pound  intensity  acts  on  one  unit  of  mass  the  acceleration  is 
one  foot  per  second.  Since  the  intensity  is  the  same  in  both 
cases,  its  measure,  the  product  of  the  mass  by  the  acceleration, 
is  constant ;  hence  we  have 

ilh.Xg  =^lbs.  XI (61) 

But  the  quantity  of  matter  in  the  second  case  is  the  unit  of 
mass,  and  the  unit  of  mass  therefore  weighs  g  lbs.  Hence  we 
have  for  the  weight  of  any  body 

w  =  mg (62) 

48.  The  Centre  of  Gravity, — The  Centre  of  Gravity  of  a  body 
is  that  point  through  which  the  action-line  of  the  body's  weight 
always  passes. 

Let  m  and  w  be  the  type-symbols  for  the  masses  and  weights 


GRAVITY.  31 


of  the  molecules  of  a  body,  g  the  weight  of  the  unit  mass,  and 
x,yy  z  the  co-ordinates  of  m.  Then,  since  the  action-lines  of  the 
molecular  weights  may  be  taken  parallel,  the  system  becomes  a 
system  of  parallel  forces  whose  centre,  which  is  the  centre  of 
gravity^  is  given  by  Eqs.  (48): 

*—  _  "^wx  _  2mgx 


2w  ^fng  ' 

—  __  'Swy  _  ^nigy 
^  ""    ^w   ""   2mg ' 
2wz        2mgz 


z  = 


(63) 


'2w         ^nig ' 
Assuming^  :=. g'  =  g"  =  etc.,  Eqs.  (63)  become 

—       2mx     —       2my      —       2mz  ,,  . 

^  =  -M-'  y  =  ^^'  '  =  -w-  •  •  •  (^4) 

The  point  defined  by  Eqs.  (64)  is  called  the  centre  of  mass. 
Therefore,  g  being  considered  constant,  the  centre  of  gravity  is 
at  the  centre  of  mass.  The  centre  of  gravity  is,  accurately,  a 
little  below  the  centre  of  mass,  but  in  ordinary  bodies  the  dis- 
tance between  them  is  negligible. 

From  Eqs.  (64)  we  see  that  the  product  of  the  mass  of  the  body 
by  either  co-ordinate  of  the  centre  of  mass,  referred  to  any  origin  what- 
ever,  is  equal  to  the  algebraic  sum  of  the  products  of  all  the  molecular 
masses  by  their  corresponding  co-ordinates  referred  to  the  same  origin. 
This  is  called  i\\Q  principle  of  the  centre  of  mass. 

Substituting  for  w,  m\  m'\  etc.,  their  values  in  terms  of  vol- 
ume and  density,  Eq.  (i),  we  have 

—       2vSx      —      2vSy      —      2v6z  ,     . 

If  the  body  be  homogeneous,  then  (J  =  <y'  =  d"  =  etc.,  and  Eqs. 


32 


MECHANICS  OF  SOLIDS. 


(65)  become 


—  __  '^vx^       —  _  "^tvy  ^      —  _  2vz 


V  ' 


V  ' 


(66) 


That  is,  in  homogeneous  bodies  the  centre  of  gravity  coincides 
with  the  centre  of  volume. 

When  a  body  is  of  such  a  form  that  we  may  obtain  expres- 
sions for  the  relations  between  its  surface  co-ordinates,  and  its 
density  is  a- function  of  these  co-ordinates,  we  may  write 


—       ^mx 
X  =  -^=^ — 


CxdM         CdxdV 


M 


f 


ddV 


-   ^my    fy'^    f^y'^ 


y  = 


f 

f,lM        f 


M 

\dM 


6dV 
dzdV 


M 


f 


ddV 


and  if  d  be  constant  we  shall  have 

CxdV 


V 


J 


V     ' 

zdV 


z  •=. 


(67) 


(68) 


From   these   equations   the   co-ordinates    of    the    centre    of 
gravity  can  be  found  by  integrating  between  the  limits  which 


GRA  FIT  v. 


33 


determine  the  volume,  when  the  expressions  which  enter  them 
are  integrable. 

49.  Determination  of  the  Position  of  the  Centre  of  Gravity. — 
The  magnitudes  whose  centres  of  gravity  are  to  be  found  are 
supposed  to  be  homogeneous  bodies,  lines  having  a  uniform 
cross-section  and  surfaces  uniform  thickness.  If  the  body  be 
symmetrical  with  respect  to  a  plane,  this  plane  may  be  taken  as 
xy^  and  we  iiave  z'-=.  o;  that  is,  the  centre  of  gravity  is  in  the 
plane  of  symmetry.  If  the  body  be  symmetrical  with  respect  to 
a  right  line  the  line  may  be  taken  as  the  axis  of  x,  and  we  have 
^  =  o,  2  =  o;  that  is,  the  centre  of  gravity  is  on  the  line  of  sym- 
metry. 

50.  Centre  of  Gravity  of  Lines. — The  centre  of  gravity  of  a  right 
line  is,  by  the  principle  of  symmetry,  at  its  middle  point. 

The  centre  of  gravity  of  broken  lines  can  be  found  by  Eqs.  ((i-^ 
when  Xy  y,  z  are  the  type-symbols  of  the  co-ordinates  of  the 
centre  of  gravity  of  each  straight  portion,  and  its  weight  w  is 
taken  proportional  to  its  length.  In  this  way  the  centre  of 
gravity  of  the  perimeter  of  any  polygon,  or  of  any  number  of 
connected  or  disconnected  right  lines,  can  readily  be  found. 

The  differential  of  any  line  is 

dl=z  Vdx*  +  d/  4-  dz*\      .-.  /  =  y*  Vdx*  +  ^'  +  dz\     (69) 
and  therefore  Eqs.  (68)  become,  for  lines, 


.£' 

Vdx'  +  dy^  +  dz* 

./> 

I 

Vdx^  -{-df-\-  dz* 

J  — 

./:■ 

I 

z  ~ 

Vdx*  +  d/  +  dz* 

.  (70) 


34 


MECHANICS  OF  SOLIDS. 


If  the  line  be  a  plane  curve  we  may  assume  xy  as  its  plane, 
and  the  above  reduce  to 


co^-^ 


y  = 


^  (71) 


2  =  0. 

M 

Ex.  I.  A  Circular  Arc. — Take  the  axis  oiy,  Fig.  18,  as  the  axis 
X  of  symmetry,  and  the  origin  of  co-ordinates  at  the 

Y  centre  of  the  arc.     Then  we  have 


y  =  ^^±j^ -^ -; 

dy  _       X  ^ 

dx"       7' 


i/^qr^«  =  ^.|/7Tl  =  ^:^l/rTf  =  y^^ 


-.-X 


y- 


dx^ 
rdx  f 


Hence  the  centre  of  gravity  of  a  circular  arc  is  on  its  radius 
of  symmetry,  and  at  a  distance  from  the  centre  of  the  circle 
equal  to  a  fourth  proportional  to  the  arc,  radius  and  chord. 


GRA  VITY, 


35 


Ex.  2,  A  Cycloid, — Let  ^,  Fig.  19,  be  the  symmetrical  axis;  then 

lY 


y=^o     and    x  =  — 


xdl 


Taking  the  equation  of  the  cycloid,  /*  =  Srx, 
we  have  o 


I  =  2{2rxy     and     dl  =  (2r)ix^^dx, 


whence 


J\2r)^xkdx 


X  = 


2(2rx)i 


Fig.  xg. 


Therefore,  for  the  curve  corresponding  to  one  complete  rota- 

tion  of  the  generating  circle,  x  =  2rand  x  =  — ;  that  is,  the  cen- 

3 
tre  of  gravity  is  on  the  axis  of  symmetry  and  at  a  distance  from 
the  vertex  equal  to  one  third   the  diameter  of  the  generating 
circle. 

51,  Centre  of  Gravity  of  Surfaces. — For  surfaces  we  have 


.  / 


xds 


X  = 


fyds 


-    f 


zds 


(7*) 


in  which  the  elementary  area  ds  is  given  by 

,         dxdy 

ds  = ^;      . 

cos  p 


(73) 


36 


MECHANICS  OF  SOLIDS. 


and  /5,  the  angle  which  the  tangent  plane  to  the  surface  makes 
with  the  plane  xyy  is  given  by 


COS  /?  =  ± 


dL 

dz 


^   dx^  "^  df  "^  dz^ 


(74) 


L^f{x,y,z)  =0 


(75) 


being  the  equation  of  the  surface. 

52.  If  the  surface  biplane,  we  may  take  it  in  the  plane  ^  and 
then  we  have 


ds  =  dx  dy, 


(76) 


and  Eqs.  (72)  become 


^        I    I  X  dx  dy 


X  = 


__    J  fy  ^y  ^^ 


y  - 


z  —  o. 


(77) 


Integrating  with  respect  to  y,  between  the  limits  /  and  /', 
these  become 


-  £Uy"-y>'^^ 


X  = 


(78) 


GRAVITY. 


37 


Ex.  I.  A  Triangle. — If  we  consider  the  area  of  the  triangle, 
Fig.  20,  to  be  made  up  of  right  lines  drawn 
parallel  to  the  base  ab^  the  weights  of  these 
lines  act  at  their  middle  points,  and  the  centre 
of  this  system  of  parallel  forces  is  somewhere 
on  the  line  cd,  drawn  from  c  to  the  middle  ^^ 
point  of  ab.  Similarly  the  centre  of  gravity 
of  the  triangle  will  be  found  on  the  lines  ae 
and  ^/ drawn  from  the  other  two  vertices  to  the  middle  points  of 
their  opposite  sides.  Hence  o  is  the  centre  of  gravity  of  the  tri- 
angle. But  we  know  from  geometry  that  the  point  of  intersec- 
tion o  is  at  two  thirds  the  distance  from  either  vertex  to  the 
middle  of  the  opposite  side.     To  show  this  analytically,  let 


aXf 


y  =  fix. 


be  the  equations  of  ac  and  ab^  Fig.  21,  the  axis  oiy  being  taken 
parallel  to  the  side  bc\  then,  Eqs.  (78), 


X^^^-p)^'^^ 


f/{a-P)xdx        3 
'^  i  f^  (a' - /r)x' dx  ,      ,    ^, 

J^     (a-fi)xdx  3 


but 


•i '—^-^x'  =  mn     and     -  ^^ ^—^-^x'  =  np  ^  -mn, 

2  32^3 


It  is  evident  that  a  straight  rod,  in  which  the  weight  of  each 
cross-section  varies  directly  as  its  distance  from  one  extremity, 
is  a  precisely  similar  problem  to  that  of  the  triangle;  and  hence 


38  MECHANICS   OF  SOLIDS. 

its  centre  of  gravity  is  at  two  thirds  its  length  from  this  ex- 
tremity. Similarly  the  areas  of  the  bounding  surfaces  of  the 
sections  of  a  cone  or  pyramid  vary  proportionally  with  their 
distances  from  the  vertex;  hence  the  centre  of  gravity  of  the 
surface  of  a  cone  or  pyramid  is  at  two  thirds  the  distance  of  the 
base  from  the  vertex.  ^ 

Ex.  2.  A  Polygonal  Plane  Area. — Divide  the  area  into  triangles, 
and  find  their  centres  of  gravity  separately;  then,  by  means  of 
the  equations 


—  __  2wy' 


(79) 


in  which  w  is  the  type-symbol  of  the  weight  of  each  triangular 
area,  and  x\  y'  the  co-ordinates  of  its  centre  of  gravity,  the 
centre  of  gravity  of  the  whole  area  may  be  found. 

Ex.  3.  A  Circular  Sector. — A  circular  sector  may  be  con- 
sidered as  made  up  of  an  indefinitely  great  number  of  equal  tri- 
angles having  equal  altitudes  and  bases,  their  vertices  being 
at  the  centre  of  the  circle.  Their  centres  of  gravity  will  be  found 
on  the  arc  of  a  circle  drawn  with  a  radius  equal  to  two  thirds 
that  of  the  given  circle.  If  the  mass  of  each  triangle  be  supposed 
concentrated  in  its  centre  of  gravity,  the  locus  of  these  masses 
will  be  a  homogeneous  line,  and  its  centre  of  gravity  will  coin- 
cide with  that  of  the  sector.  Therefore,  calling  r  the  radius  of 
the  circular  sector,  c  its  chord,  and  a  the  length  of  its  arc,  we 
have,  taking  the  axis  x  to  be  the  axis  of  symmetry, 

—  __2  re 

That  is,  the  centre  of  gravity  of  a  circular  sector  is  on  its  ra- 
dius of  symmetry,  and  at  a  distance  from  the  centre  equal  to 
two  thirds  of  a  fourth  proportional  to  the  arc,  radius  and  chord. 


GRA  VITY. 


39 


Ex.  4.  A  Circular  Segment, — Let  the  origin  be  at  the  centre 
of  the  circle,  Fig.  22,  and  the  axis  of  x  that  of  symmetry  ;  then 


l{y"-y)xdx 


X  = 


;  .    .    .    (80) 


y-o\  x"  +/  =  r\ 

Then  we  have 

and  Eq.  (80)  becomes 

2  r^(r'  -  x')^x  dx       i(r«  -  x'')^ 


x  = 


But  AB  -  2(r'  -  x'')^  =  e;  .'.  x  =  —, 

12S 


Therefore  the  centre  of  gravity  of  a  circular  segment  is  on  the 
radius  drawn  to  the  middle  of  the  arc,  and  at  a  distance  from 
the  centre  equal  to  the  cube  of  the  chord  divided  by  twelve  times 
the  area  of  the  segment. 

Ex.  5.  An  Area  bounded  by  a  Parabola^  its 
Axis  and  one  of  its  Ordinates. — Take  the  para-  c 
bola  as  in  Fig.  23,  and  we  have 


/"•  =  2^X', 


o; 


t/o «/o 

t/o  t/o 


Fig.  23. 


=  iy': 


40  MECHANICS  OF  SOLIDS. 

^r'\y"-y')dx     i^r"2pxdx 
jf    {y'-y')dx        £    ^px'-dx 

For  the  parabolic  spandrel  OBC  we  have 
y'  =  ^;        .'  =  o; 

These  latter  values  can  be  readily  determined  by  the  appli- 
cation of  the  principle  of  moments  ;  thus,  the  area 

OBA  =  iOCBA,     and     OBC  =  iOCBA, 

The  sum  of  the  moments  of  the  weights  of  OBA  and  OBC 
with  respect  to  O  must  be  equal  to  the  moment  of  the  weight  of 
the  rectangle  with  respect  to  the  same  point.     Hence  we  have 

3  5  3  2 

and  therefore 

-      3^" 


for  the  X  co-ordinate  of  the  centre  of  gravity  of  the  parabolic 


GRAVITY.  41 


spandrel.  Th& y  co-ordinate  is  obtained  by  considering  the  fig- 
ure to  be  resting  on  OC,  and  taking  the  moments  again  with  re- 
spect to  O,     We  have 


2W      s/'    ,    PV-_  Wy" 
3"^     8     "^    3-^~     2    ' 


whence 


4 


Ex.  6.  An  Elliptic  Quadrant. — The  equation  of  the  ellipse  re- 
ferred to  its  centre  and  axes  is  a^y^  +  b*x^  =  a'^';  whence  we 
have 


/' =i(fl' -  ;.')*;     /  =  °; 

r—(a*  —  x^yx  dx  ,       . 

X  =  — f ^ =  J^J-  ±(a'  -  M'  =  ^; 

Trad  nab\     sa^  ^  Jo       ^Tt 

4 


•^  Ttab  37r* 

4 

If  d5  =  <^  the  ellipse  becomes  a  circle,  and  the  co-ordinates  of 
the  centre  of  gravity  of  a  circular  quadrant  referred  to  its  centre 
are 

-      -      4r 
x=y  =  — . 
37r 

A, 

53.  Surfaces  of  Revolution. — Let  x  be  the  axis  of  revolution; 
and  since  it  is  an  axis  of  symmetry,^  and  z  are  both  zero.  Let 
y=-f{x)  be  the  equation  of  the  curve  whose   revolution   gener- 


42  MECHANICS  OF  SOLIDS. 

ates  the  surface,  and  we  have  for  the  elementary  surface 


hence 


X  = 


C^  27tyx{dx'  +  df)^         j^yxidx"  +  df)^ 
r  27ty{dx^  +  df)^  ry{dx^  +  df)^ 


Ex.   I.    A    Spherical    Zone, — The    equation   of    the  circle  is 
^'  -[.y  =  r";  then  we  have 


ryx(dx^  ■\'  dy^^         f    rx  dx         , 


+  x' 


r  y(dx?  +  d/f  r  rdx 

Jx"  J x" 


Whence  the  centre  of  gravity  of  a  spherical  zone  is  at  the  middle 
point  of  the  right  line  joining  the  centres  of  its  bases. 

Ex.  2.  A  Right  Conical  Surface. — Let  the  equation  of  its  gen- 
erating line  be  J'  =  ax^  and  we  have 


dx 

X  =  ; =  —X\ 


X  dx 


Hence  the  centre  of  gravity  of  a  right  conical  surface  is  on 
its  axis  and  two  thirds  of  the  distance  from  the  vertex  to  the 
base  ;  it  is  independent  of  the  angle  of  the  cone,  and  hence  is  a 
common  point  for  all  right  cones  having  the  same  vertex,  axis 
and  altitude. 

54.  Centre  of  Gravity  of  Volumes. — For  volumes  we  have,  in 
general,  dv  =  dxdy  dz,  and  Eqs.  (68)  become 


GRAVITY. 


43 


/    /    j  X  dx  dy  dz 


X  = 


/    /    I  ^  ^^  dy  dz 
/    /    I  z  dx  dy  dz 


(8.) 


55^  Volumes  of  Revolution. — A  volume  of  revolution  being  sym- 
metrical with  respect  to  its  axis,  if  we  take  this  axis  as  the  axis 
of  x  we  shall  have,  Eqs.  (8i), 


/xdV         I     nfxdx        fy'^xdx 
_  Jjc" _  Jx" . 


X  = 

7=o; 
2=0. 


(82> 


Ex.  I.  A  Paraboloid  of  Revolution. —  To  find  the  centre  of 
gravity  of  a  portion  of  a  paraboloid  of  revolution  limited  by 
planes  perpendicular  to  the  axis,  we  have  for  the  equation  of  the 
generating  curve 

/  =  2px, 
and  for  the  co-ordinate  of  the  centre  of  gravity 


r2pX* 
X^—. 


dx 


r 


2px  dx 


_   2(X''  -  X"') 


44  MECHANICS  OF  SOLIDS. 

When  ^"  =  o  we  have 


X  =  4x 


Ex.  2.  A  Spheroid. — Taking  the  origin  at  the  extremity  of 
the  axis,  the  equation  of  the  generating  curve  is 

and  for  a  portion  of  the  spheroid  from  the  origin  to  x'  we  have 

For  half  of  the  spheroid  we  have  x'  =  ^,  and 

x  =  ^a\ 

which  is  independent  of  the  shape  of  the  spheroid. 

56.  Whenever  the  volumes  whose  centres  of  gravity  are  to  be 
found  are  such  that  we  can  connect  the  areas  of  their  successive 
sections  normal  to  any  line  by  some  law,  their  centres  of  gravity 
can  be  found  from  the  general  equations  by  a  single  integration. 
Thus  to  find  the  centre  of  gravity  of  any  cone  or  pyramid,  first 
find  the  centre  of  gravity  of  its  base  and  join  this  point  with  the 
vertex.  It  is  evident  that  the  line  so  drawn  will  pierce  the  suc- 
cessive sections  in  their  centres  of  gravity,  and  at  these  points 
the  weights  of  the  several  sections  will  act,  with  intensities 
which  are  proportional  to  their  areas.  Then  Eqs.  (67),  when  X 
is  the  area  of  any  section  parallel  to  the  base,  become 


Jx" 


Xx  dx 

-; (83) 


GRAVITY.  45 


the  line  from  the  vertex  perpendicular  to  the  base  being  taken 
as  the  axis  of  x. 

Ex.  A  Pyramid  or  Cone. — Take  the  origin  at  the  vertex,  and 
let  A  be  the  area  of  the  base  and  x'  the  abscissa  of  its  centre  of 
gravity.     Then  we  have  for  any  section 


^-^^, 


and 


-  #r- 


t/o 


*  =  V^p —  =  -*'■ 


Therefore  the  centre  of  gravity  of  a  pyramid  or  cone  is  on 
the  line  joining  the  vertex  with  the  centre  of  gravity  of  the  base, 
and  at  a  distance  from  the  vertex  equal  to  three  fourths  of  its 
length. 

57.  Theorems  of  Pappus. — Clearing  the  second  of  Eqs.  (71) 
and  (78)  of  fractions  and  multiplying  both  members  of  each  by 
2;r,  we  have 

27tyl  =  j2nydl. (84) 

2nys  =  J^7t(y'' ^  y')dx (85) 

The  second  member  of  Eq.  (84)  is  the  expression  for  the  area 
of  the  surface  of  revolution  generated  by  the  curve  /  about  the 
axis  X,  and  that  of  Eq.  (85)  is  the  expression  for  the  volume  gen- 
erated by  the  revolution  of  the  plane  surface  s  about  the  same 
axis.  Hence  we  have  by  these  equations  a  simple  means  of  de- 
termining  an  area  or  volume   of  revolution  whenever  we  can 


4^  MECHANICS   OF  SOLIDS. 

find  the  position  of  the  centre  of  gravity  of  the  generating  line 
or  surface.  For  by  the  first  members  we  see  that  such  area  or 
volume  is  equal  to  the  product  obtained  by  multiplying  the 
length  of  the  generating  line,  or  the  area  of  the  generating  plane 
surface,  by  the  circumference  described  by  its  centre  of  gravity. 
These  theorems  are  useful  in  mensuration. 


Graphical  Statics. 

58.  Roof  and  bridge  trusses  are  usually  framed  structures 
composed  of  beams  united  by  rods  and  struts,  and  are  subjected 
to  certain  stresses  due  to  the  weights  of  the  assembled  parts,  the 
loads  they  are  required  to  support,  and  wind  pressure.  While 
the  computation  of  these  stresses  can  be  made  by  the  usual  ana- 
lytical methods,  the  processes  of  graphical  statics  are  so  simple 
and  accurate  as  to  make  them  of  frequent  application.  The  fol- 
lowing pages  contain  merely  an  exposition  of  its  simplest  fun- 
damental principles,  to  which  are  added  a  few  of  the  more  ele- 
mentary illustrative  examples.*  The  further  development  more 
properly  belongs  to  applied  mechanics  in  Civil  Engineering. 

59.  Reciprocal  Figures, — Any  two  figures  are  said  to  be  recipro- 
cal when  the  first  can  be  derived  from  the  second  in  the  same 
way  that  the  second  is  obtained  from  the  first.  Reciprocal  fig- 
ures applicable  to  graphical  statics  are  subject  to  the  following 
conditions  : 

(i)  The  sides  of  one  should  be  respectively  perpendicular  or 
parallel  to  those  of  the  other. 

(2)  Lines  radiating  from  a  vertex  in  one  figure  should  be  per- 
pendicular or  parallel  to  corresponding  sides  forming  a  closed 
polygon  in  the  other. 

(3)  Each  figure  should  be  composed  of  the  same  number  of 
closed  polygons,  and  each  line  of  the  figure  should  make  a  part 
of  two  of  these  polygons,  and  of  two  only. 

*  La  Statique  Graphique  et  ses  applications  aux  construction,  par  Maurice 
Levy.     Gauthier-Villars;  Paris,  1874. 


GRAPHICAL    STATICS.  4/ 

(4)  At  least  three  lines  should  meet  at  each  vertex,  and  each 
side  should  pass  through  at  least  two  vertices. 

The  two  figures  D  A  B  C  and  dab  c  (Fig.  24),  each  formed  by 
the  six  lines  joining  four  points  in  ^ 

■a  plane,  fulfil  all  the  above  condi-  /\\ 

tions,  and  are  therefore  reciprocal       ^O-^^o^CX 
figures  applicable  to  graphical  stat-  B^— — ^ — — ^ 
ics.    Either  of  these  figures  is  com-  Fig.  24. 

pletely  determined  when  any  five  of  its  six  lines  are  given  ;  for, 
any  five  of  the  lines  form  two  triangles  having  a  common  side, 
and  the  sixth  line  joins  two  vertices  which  are  already  fixed  in 
position  by  the  five  given  lines.  Hence  if  all  the  lines  in  two 
figures,  except  one  in  each,  are  known  to  fulfil  the  conditions  of 
reciprocal  figures,  the  figures  are  reciprocal. 

60.  The  Force  Polygon. — Let  there  be  a  given  system  of  co- 
planar  forces  ;  then  if,  from  any  point  in  the  plane,  a  polygon 
be  constructed  whose  sides  taken  in  order  represent  the  direc- 
tions and  intensities  of  the  several  forces  of  the  system,  this 
polygon  is  called  2i  force  polygon  of  the  system.  By  the  principle 
of  the  parallelogram  of  forces  we  readily  see  that,  if  the  polygon 
be  closed,  the  resultant  of  the  system  is  zero  and  the  forces  are 
in  equilibrio  ;  and  in  general,  that  the  right  line  required  to 
close  the  polygon,  when  reversed,  represents  the  resultant  of  the 
system  in  intensity  and  direction, 

61.  The  Polar  Polygon. — Let  the  polygon  abcde,  Fig.  25,  be  the 
force  polygon  of  the  system  of  forces  i  i',  2  2',  3  3',  4  4',  Fig. 
26.  Assume  any  point  O  in  its  plane  as  a  pole,  and  draw  the 
lines  Oa,  Ob,  Oc,  etc.,  to  its  vertices.  In  the  plane  of  the  forces 
draw  any  line  i?^"  parallel  to  Oa  of  the  force  polygon,  that  is, 
parallel  to  the  line  joining  the  pole  and  the  origin  of  the  side  i 
of  the  force  polygon,  and  mark  its  intersection  with  the  force 
I  i'  by  the  symbol  i.  From  i  draw  i  2  parallel  to  Ob,  and  mark 
its  intersection  with  the  force  2  2'  by  2.  In  the  same  way  fix  the 
points  3,  4  and  P;  the  latter  point  being  the  intersection  of  the 
line  drawn  through  4  parallel  to  Oe  with  the  first  line  drawn. 
The  polygon  i  2  3  4  i?i  is  called  a  funicular  ox  polar  polygon  of  the 


48 


MECHANICS  OF  SOLIDS. 


system  of  forces,  with  reference  to  the  pole  O.  Since  i^  might 
have  occupied  any  position  parallel  to  Oa,  it  is  determinate  in 
direction  but  arbitrary  in  position;  therefore  the  polygon  con- 
structed is  but  one  of  an  indefinitely  great  number  of  polar 


Fig.  96. 


polygons  belonging  to  the  pole  O,  all  of  which,  however,  having 
parallel  sides,  are  similar  figures.  Assuming  another  pole,  (?', 
and  line  R'R"  parallel  to  O'a,  the  polar  polygon  i'  2'  3'  4'  H'^ 
may  be  constructed,  belonging  to  another  set  of  similar  poly^ 
gons;  and  so  on  indefinitely.     From  their  construction  it  is  evi- 


GRAPHICAL   STATICS.  49 

dent  that  the  force  polygon  and  the  polar  polygon  are  reciprocal 
figures. 

62.  Properties  of  Polar  Polygons. — The  following  properties  of 
polar  polygons  are  of  use  in  graphical  solutions: 

(i)  The  intersections  of  the  corresponding  sides  of  two  polar  poly- 
gons belonging  to  the  same  system  of  forces  are  on  a  right  line  parallel 
to  (he  line  joining  the  two  poles.  To  show  this,  produce  the  corre- 
sponding sides  of  the  two  polar  polygons,  i  2  and  i'  2',  2  3  and 
2'  3',  etc.,  until  they  intersect;  the  points  i",  2",  3",  4",  ^" 
will  lie  on  the  same  right  line  Ofi^  drawn  through  the  point  of 
intersection  i?"  of  iR  and  i'R\  and  parallel  to  00'\  then  five  of 
the  six  lines  joining  the  points  R'\  i,  i'  and  i",  viz.,  i  i',  i  R*', 
I  i",  1'  R'\  \'  i",  of  the  polar  polygon  are  respectively  parallel 
to  the  five  lines  ab,  Oa,  Ob,  0*a,  0*b,  joining  the  four  points  (9, 
0\  a,  b,  of  the  force  polygon;  therefore  the  sixth  lines,  ^"i"  and 
00',  of  these  reciprocal  figures  are  also  parallel.  In  the  same 
way  the  other  points,  2",  3",  4",  may  be  shown  to  be  on  the 
right  line  through  R"  parallel  .to  00'.  Hence  the  change  from 
the  pole  O  to  O'  is  equivalent  to  supposing  that  each  of  the  sides 
of  the  first  polar  polygon  rotates  around  each  point  of  intersec- 
tion until  it  coincides  with  the  corresponding  side  of  the  second 
poliir  polygon. 

As  the  pole  O'  approaches  O,  the  vertices  i  and  i'  of  the  two 
polar  polygons  remaining  fixed,  the  line  0^0^  moves  toward  in- 
finity, always  remaining  parallel  to  OO'.  When  O'  coincides 
with  O,  the  sides  of  the  polar  polygons  become  parallel  and  they 
become  two  of  the  same  set,  and  Ofi^  is  at  an  infinite  distance. 
Hence  the  parallelism  of  polar  polygons  relative  to  the  same 
pole  is  merely  a  particular  case  of  polar  polygons  relative  to 
different  poles. 

(2)  If  a  system  of  forces  and  one  of  its  polar  polygons  be  given, 
every  other  polar  polygon  of  the  system  may  be  constructed  without  the 
aid  of  the  force  polygon.  Thus,  suppose  that  the  one  relative  to 
O  be  known.  Draw  the  arbitrary  line  O fi ^^  and  prolong  the 
sides  of  the  known  polygon  to  meet  this  line  at  \" ,  2",  3",  etc.; 
then  draw  through  any  point  thus  determined,  as  R'\  a  line  ar' 
4 


50  MECHANICS   OF  SOLIDS. 

bitrary  in  direction  meeting  the  force  i  i'  in  i';  from  i'  draw  i' 
i",  and  where  it  meets  the  force  2  2'  will  be  the  vertex  2';  from 
2'  the  line  2'  2"  will  determine  3'  by  its  intersection  with  the 
force  3  3',  and  so  on.  To  show  that  this  polygon  i'  2' 3' 4' 7?'  i' 
is  a  polar  polygon,  draw  through  the  origins  of  the.  force  poly- 
gon the  right  line  aO'  parallel  to  t* R" ,  and  through  O  a  parallel 
to  O^  O^,  and  let  (9'  be  their  point  of  intersection.  The  two  fig- 
ures formed  by  five  of  the  lines  joining  the  four  points  i?",  i,  i', 
i",  and  O,  0\  «,  b  are  parallel  and  reciprocal,  and  therefore  the 
sixth  lines  i'  i"  and  O'  b  are  parallel  and  the  figures  are  recipro- 
cal. In  the  same  way,  taking  the  figures  corresponding  to  the 
four  points  2,  2',  i",  2",  and  (9,  0\  b,  c,  it  can  be  shown  that 
2'  2"  and  O*  c  are  parallel  and  reciprocal,  and  so  on  ;  hence  the 
polygon  1'2'3'4'i?'  i',  being  reciprocal  to  the  force  polygon,  is 
a  polar  polygon  with  reference  to  the  pole  O'. 

(3)  The  intersection  of  any  two  sides  of  the  polar  polygoti  is  a  point 
of  the  resultant  of  the  forces  represented  by  the  lines  included  between 
the  corresponding  vertices  of  the  force  polygon.  Prolong  the  sides 
i'^'  and  2'  3'  till  they  meet  at  ^',  and  the  forces  i  and  2  till  they 
intersect  at  c'\  The  figures  formed  by  joining  O' ,a,  b,  <r,  and  i', 
2',y,  ^",  are  reciprocal  ;  and  hence  c^  c"  is  the  action-line  of  the 
resultant  of  i'  and  2',  since  it  is  parallel  to  acand  passes  through 
^".  Similarly^'  d''  is  the  action-line  of  the  resultant  of  ac  and  3, 
or  of  I,  2  and  3,  and  RE'  is  the  action-line  of  the  resultant  of 
the  whole  system. 

Hence  to  find  the  resultant  of  the  whole  or  any  part  of  any 
system  of  co-planar  forces  by  graphical  construction,  draw  any 
polar  polygon  on  the  action-lines  of  the  forces  whose  resultant 
is  to  be  found;  the  intersection  of  the  extreme  sides  will  be  a 
point  of  the  resultant,  and  a  line  drawn  through  this  point  paral- 
lel to  the  closing  line  of  the  force  polygon  of  the  forces  in  ques- 
tion will  be  the  action-line  of  the  resultant.  The  resultant  is 
then  completely  determined  by  laying  off  on  this  line,  in  the 
proper  direction,  a  distance  equal  to  the  length  of  the  closing 
line  of  the  force  polygon. 

(4)  When  a  system  of  co-planar  forces  is  in  equilibrio^  and  the  posi- 


GRAPHICAL  STATICS. 


51 


tions  of  the  action- lines  of  three  unknown  forces  are  given^  the  intensities 
of  these  unknown  forces  may  be  determined  by  means  of  the  polar  poly- 
gon. Let  the  known  forces  i,  2,  3  and  4,  and  the  unknown  forces 
5,  6  and  7,  Fig,  27,  be  a  system  in  equilibrio.  Suppose  the 
known  forces  to  be  in  equilibrio  with  two  of  the  unknown 
forces,  as  5  and  7,  and  construct  the  force  polygon  a  b  c  d  e  g  a. 
Assume  any  pole,  as  Oy  draw  Oa^  Ob^  Ocy  Oe  and  Og^  and  construct 


Fig. 


the  polar  polygon  7'  i'  2'  3'  4'  5'.  The  remaining  lines  of  the 
polar  polygon  of  the  whole  system  must  radiate  from  5'  and  7' 
and  intersect  on  the  action-line  of  the  force  6.  To  determine 
these  lines  we  must  construct  a  figure  which  shall  be  reciprocal 
to  that  formed  by  joining  O,  g,  and  the  two  extremities  of  6  in 
the  force  polygon.  Two  vertices  of  the  required  figure  are  5'  and 
n.  The  lines  nm,  mt/ ^  and  ^'n  are  reciprocal  to  those  radiating 
from  gy  and  m  is  therefore  another  vertex  of  the  required  figure. 


52  MECHANICS  OF  SOLIDS. 

Join  7'  and  ;«,  and  the  intersection  of  this  line  with  the  action- 
line  of  the  force  6  gives  the  remaining  vertex.  The  required 
figure  is  therefore  w«5',6'.  Now  complete  the  reciprocal  of  this 
figure  in  the  force  polygon  by  drawing  Of  parallel  to  5'6',  and 
Oh  parallel  to  7'6'.  Then  draw  /z/,  which  is  parallel  to  nd'  by 
the  reciprocity  of  the  figures,  and  we  have  5,  6  and  7  in  the  force 
polygon  to  represent  the  intensities  and  directions  of  the  forces 
which  were  to  be  determined. 

63.  Problems. — Since  the  triangle  is  the  only  polygonal  figure 
that  cannot  change  its  form  without  changing  the  length  of  one 
of  its  sides,  it  is  made  the  basis  of  all  frame-work.  In  the  frames 
here  discussed  the  parts  are  supposed  to  be  free  to  rotate  about 
the  joints  at  the  vertices  of  the  frame,  and  they  are  therefore 
subjected  to  longitudinal  strains  only. 

The  foregoing  principles  enable  us  to  find  the  stresses  on 
the  parts  of  a  frame  which  is  a  plane  figure,  when  subjected  to 
the  action  of  forces  which  are  co-planar  with  it.  Two  diagrams 
are  constructed,  one,  called  the  frame  diagram^  to  represent  the 
frame  and  the  action-lines  of  the  forces,  and  the  other,  the  strain 
diagram,  to  represent  the  force  polygon  and  the  stresses  on  the 
various  parts. 

To  facilitate  the  construction  and  reading  of  the  strain  dia- 
gram the  following  notation  is  employed:  In  the  frame  diagram 
each  triangular  space  is  marked  by  a  letter,  and  exterior  to  the 
frame  each  space  bounded  by  the  action-lines  of  adjacent  forces 
is  also  thus  marked.  Thus,  in  Fig.  28,  A  designates  the  left- 
hand  space  of  the  frame,  H  the  exterior  space  between  forces 
I  and  8,  and  O  the  exterior  space  between  the  forces  7  and  8. 
Any  line  or  vertex  in  the  figure  is  designated  by  the  letters  of 
the  spaces  separated  by  it.  Thus,  HO  designates  the  force  8, 
HI  the  force  i,  and  HA  the  line  of  the  frame  between  the  spaces 
H  and  A.  The  vertex  at  which  force  i  acts  may  be  designated 
by  I A  or  HB.  HO  designates  both  a  force  and  a  vertex.  In 
such  a  case  one  is  called  the  force  HO,  and  the  other  the  vertex  HO^ 

In  the  strain  diagram,  Fig.  29,  the  vertices  are  lettered,  and 
any  line  of  the  diagram   is  designated  by  the  letters  at  its  ex^ 


GRAPHICAL    STATICS.  53 

tremities.  The  letters  are  so  arranged  in  the  two  diagrams 
that  reciprocal  lines  shall  be  designated  by  the  same  letters. 
This  arrangement  is  shown  in  the  figures. 

(i)  The  frame  represented  in  Fig.  28  is  subjected  to  the 
action  of  the  forces  i,  2,  3,  4,  6  and  7,  as  indicated  by  their 
action-lines,  the  extreme  vertices  HO  and  LM  being  points  of 
support.  It  is  required  to  find  the  stresses  on  the  parts  of  the 
frame. 

To  find  the  reactions  at  the  points  of  support,  construct  the 
force  polygon  HIJKLNO,  Fig.  30.  HO  represents  the  intensity 
of  the  resultant  of  the  applied  forces.  To  find  its  point  of  appli- 
cation, assume  the  pole  (9',  Fig.  30,  and  construct  the  polar 
polygon  i?'i'2'3V6'7'^',  Fig.  28.  R*  is  then  the  point  of  ap- 
plication of  the  resultant,  and  its  action-line  is  therefore  R' R. 
Through  the  points  of  support  draw  5  and  8  parallel  to  R'R, 
and  divide  HO,  Fig.  30,  into  two  parts  which  shall  be  to  each 
other  as  the  distances  of  R*  from  the  action-lines  of  5  and  8. 
Thus  making  OP  equal  to  the  distance  between  the  forces  5  and 
8,  and  OQ  equal  to  the  distance  of  R'  from  8,  and  drawing  QR 
parallel  to  PH,  we  have  OR  as  the  intensity  of  force  5,  and  RH 
that  of  force  8. 

Now  construct  the  force  polygon  of  the  whole  system 
HIJKLMNOH,  Fig.  29,  LM  being  the  intensity  of  5,  and  OH 
that  of  8. 

To  construct  the  strain  diagram  begin  with  a  vertex  where 
only  two  forces  are  unknown,  as  the  point  of  support  HO.  On 
HO^  Fig.  29,  construct  the  reciprocal  of  this  vertex  by  drawing 
through  ZTa  line  parallel  to  HA,  Fig.  28,  and  through  O  a  line 
parallel  to  AO.  The  triangle  OHAO  gives  the  intensities  of  the 
three  forces  acting  at  the  vertex  HO.  Since  OH  is  the  direction 
of  the  force  8,  the  directions  of  the  other  two  are  HA  and  AO. 
Also,  since  HA  acts  along  the  frame-piece  towards  the  vertex  it 
produces  cofnpression,  and  since  AO  2iCts  from  the  vertex  the  latter 
acts  as  a  tension. 

Passing  now  to  the  vertex  HB,  we  have  HA  and  -^/ known, 
and  drawing  AB  and  IB,  Fig.  29,  parallel  to  AB  and  IB,  Fig. 


54 


MECHANICS  OF  SOLIDS. 


Fig.  a8. 


Fig.  29. 


GRAPHICAL   STATICS. 


55 


2S>,  we  have  the  reciprocal  of  the  vertex  IIB,  which  is  AHIBA^ 
giving  the  stresses  on  the  pieces  which  meet  at  that  vertex. 
The  direction  of  HI  is  known,  and  we  see  by  following  the 

1 


Fig. 


30. 


strain  polygon  that  IB^  BA  and  AH  all  act  towards  the  vertex. 
The  stresses  on  the  corresponding  pieces  are  therefore  com- 
pressions. 

For  the  vertex  AC  we  have  in   the  strain  diagram  NO^  OA 


5^  MECHANICS  OF  SOLIDS. 

and  AB,  Draw  BC  and  NC  parallel  to  the  corresponding  lines 
in  the  frame,  and  we  have  NOABCN  ior  the  reciprocal  of  the 
vertex  AC.     BC  is  compression  and  CN  tension. 

Taking  the  remaining  vertices  in  succession,  we  have: 

¥or  B/y  the  polygon  CBIJDC]  JD  being  compression  and 
Z>C  tension. 

For  iV^^,  the  polygon  MNCDEM  \  DE  and  EM  being  ten- 
sions. 

For  DK^  the  polygon  EDJKFE\  KF  being  compression  and 
FE  tension. 

For  EG,  the  polygon  MEFGM  \  FG  and  GM  being  tensions. 

For  FL,  the  polygon  GFKLG\  LG  being  compression. 

For  LM^  the  polygon  LMG. 

From  which  all  the  stresses  and  their  characters  are  com- 
pletely determined. 

(2)  The  Simple  Warren  Truss,  having  equal  loads  at  the  lower 
vertices  (Fig.  31). 

Construct  the  force  polygon  of  the  applied  forces  IJKLM, 
Fig.  32.  Their  resultant  is  /J/",  which  evidently  acts  through 
the  middle  point  of  the  truss.  The  reactions  at  the  points  of 
support  are  each  equal  to  half  the  total  load,  and  are  MA  and  AI. 

For  the  vertex  AI  we  have  the  polygon  A  IB  A  ;  IB  being 
tension  and  BA  compression. 

For  the  vertex  IC  we  have  BIJCB\  JC  being  tension  and 
BC  compression. 

For  the  vertex  AC  ^t.  have  ABCDA\  CD  being  tension  and 
DA  compression. 

For  the  vertex  Df  we  have  DCJKED\  and  for  the  vertex 
AE  we  hsLVQ  ADEFA.  In  the  last  two  polygons,  since  E  and  Z>, 
and  ^  and  Fare  coincident,  the  parts  DE  and  EF  support  no 
stress. 

The  stresses  on  the  remaining  parts  may  be  found  by  pro- 
ceeding with  this  construction  through  the  vertices  H^F,  AG  and 
ZZT,  or  by  a  construction  similar  to  the  above,  beginning  at  the 
vertex  AM. 

The  upper  chord  is  subjected  to  a  stress  of  compression  and 


GRAPHICAL   STATICS. 


57 


the  lower  chord  to  one  of  tension,  these  stresses  being  greatest 
at  the  middle  of  the  truss,  while  the  stresses  on  the  diagonals 
are  greatest  at  the  ends  of  the  truss. 

(3)  The  Simple  Warren  Truss,  loaded  unequally  at  the  lower 
vertices. 

Let  the  frame  be  the  same  as  in  the  preceding  example,  Fig. 

Fig.  31. 


Fig.  32. 


Fig.  33. 


31,  and  let  the  two  right-hand  loads  be  each  one  half  as  great  as 
the  others.  The  force  polygon  of  the  applied  forces  is  IJKLM 
{Fig.  33).  The  intensity  of  their  resultant  is  IM.  To  find  its 
action-line  and  the  reactions  at  the  points  of  support,  assume 
the  pole  O  and  construct  the  polar  polygon  6123456.    -ffis  the 


58 


MECHANICS  OF  SOLIDS. 


action-line  of  the  resultant,  and  drawing  OA  parallel  to  5  6,  we 
have  MA  and  AI  as  the  intensities  of  the  reactions. 

The  determination  of  the  strains  on  the  parts  is  sufficiently- 
indicated  by  the  nomenclature  in  the  diagram. 


Fig.  34. 


(4)  Figs.  34,  35,  36  and  37  represent  loaded  trusses  with  their 
strain  diagrams.  As  an  exercise  the  student  should  supply  the 
nomenclature  and  determine  the  character  of  the  strains.  It 
will  be  observed  that  at  one  vertex  of  the  frame  in  Fig.  37  there 
are  three  forces  to  determine,  and  the  problem  therefore  requires 
the  application  of  proposition  4,  Art.  62. 


GRAPHICAL   STATICS. 


59 


6o 


MECHANICS  OF  SOLIDS. 


. 

/} 

\\ 

■< 

'),. , 

<^' 

2 

h 

7 

^"^"""^^           # 

iv^ 

■ . 

^^~-~>^^ 

Fig.  37. 


Work  and  Energy. 

64.  Work  is  said  to  be  done  by  a  force  when  its  point  of  ap- 
plication has  any  motion  in  the  direction  of  its  action-line.  The 
unit  of  work  is  the  quantity  of  work  done  by  a  force  of  unit 
intensity  while  its  point  of  application  moves  over  the  distance 


WORK  AND  ENERGY,  .6l 

unity  in  the  direction  of  the  action-line  of  the  force.  The  unit 
of  work  used  throughout  the  text  is  called  the  foot-pound^  the 
unit  of  distance  being  one  foot  and  the  unit  of  intensity  one 
pound.  The  unit  of  work  of  the  C.  G.  S.  system  is  the  work 
done  by  a  dyne  over  a  centimeter,  and  is  called  the  Erg, 

The  simplest  illustration  of  work  is  that  of  lifting  a  weight 
through  a  vertical  height.  Thus,  it  requires  the  expenditure  of 
one  foot-pound  of  work  to  lift  one  pound  through  a  height  of 
one  foot.  Hence,  the  work  done  in  lifting  a  weight  through 
any  height  is  equal  to  the  product  of  the  weigiit  and  height; 
and,  in  general,  the  work  done  by  any  constant  force  is  found  by 
multiplying  its  intensity  by  the  path  of  its  point  of  application, 
estimated  in  the  direction  of  its  action-line.  If  the  force  be 
variable  it  may  be  regarded  as  constant  while  its  point  of  appli- 
cation describes  a  path  dp,  estimated  along  its  action-line,  and  the 
elementary  quantity  of  work  is 

dW=Idp (86) 

The  summation  of  all  the  elementary  quantities  of  work  gives 
the  total  quantity  done  by  this  force,  and  we  have 

W^^dW=:2Idp (87) 

Where  the  principles  of  the  calculus  can  be  applied,  we  have 

W=fdW=Jldp (88) 

Hence,  whenever  such  a  relation  can  be  established  between  the 
variable  intensity  /and  the  path  /  so  that  the  second  member 
can  be  expressed  as  a  known  integrable  function  of  a  single  va- 
riable, the  total  quantity  of  work  can  be  determined  by  integra- 
tion. 


62 


MECHANICS  OF  SOLIDS. 


65.  The  symbol  expressing  work,  /  Idp,  is  analogous  to  the 

symbol   j  ydx  in  the  calculus.     The  latter  is  the  representative 

of  the  quadrature  of  a  curve  whose  varying  ordinates  are  j'  and 
abscissas  x,  both  expressed  in  the  same  unit.  From  this  analogy 
we  may  graphically  represent  work  by  the  area  contained  be- 
tween the  axis  of  p  and  a  curve  whose  ordinates  measure  the 
varying  intensity  of  the  working  force  at  the  different  points  of 
the  path  p.  The  unit  of  work  is  graphically  represented  by  a 
square  whose  side  is  the  unit  of  the  scale  from  which  the  inten- 
sity and  path  are  taken. 

When  the  expression  /  Idp  is  not  integrable,  the  quantity  of 

work  can  be  determined  approximately 
by  the  usual  methods  for  the  estima- 
tion of  the  area  included  between  the 
curve,  the  extreme  ordinates,  and  the 
path,  as  in  mensura  tion.  Thus  let  the 
ordinates  of  the  curve  j^',  Fig.  38,  repre- 
sent the  varying  intensity  of  the  force 
while    its    point    of    application    passes 

over  the   path  pp\     Poncelet's    formula   for   the   approximate 

area  is 


^ 

V- 

"^ 

I1 

u 

I3 

I4 

I7 

^^-^ 

£- 

Fig.  38. 


Q  =  d[2{/,  +  /,  +  /.  +  .../.)+  i(/_ +/.+,)_  i(/.+/«)],(89) 


in  which  d  is  the  distance  between  the  consecutive  ordinates  7^, 
/„ /g,  etc.,  when  the  whole  path,^',  is  divided  into  any  even 
number,  n,  of  equal  parts. 

If  the  varying  values  of  /  be  known  only  at  certain  points  of 
the  path,  the  extremities  of  these  may  be  joined  by  right  lines, 
thus  forming  trapezoids  whose  aggregate  area  ox\\y  approximately 
represents  the  quantity  of  work. 

66.  Energy, — Force  and  matter  are  inseparably  connected. 
Any  system  of  masses  is  accompanied  by  forces,  and  these  forces 


WORK  AND  ENERGY.  63 

perform  work  during  any  change  in  the  configuration  of  the  sys- 
tem. Energy  is  the  capacity  for  doing  worky  and  it  is  measured 
in  units  of  work.  Taking  a  single  molecule  of  the  system  mov- 
ing under  the  action  of  the  resultant  of  all  the  forces  applied  to 
it,  we  have 

Idp  =  m-^ds\      .......     {90) 

and  for  the  whole  system, 

.     2n^p=^-^^<}i-^i!^ (9.) 

t/a  2  2 

When  the  molecule  is  at  the  position  i,  the  quantity  of  work 
represented  by  the  first  member  of  Eq.  (91)  is  called  potential 
energy,  since  it  measures  the  capacity  of  the  force  to  do  work 
while  the  molecule  passes  from  i  to  2.  It  is  simply  energy  of 
position;  that  is,  by  virtue  of  the  position  of  the  molecule  in  the 
system  the  forces  acting  upon  it  have  a  certain  power  to  do  work 
while  its  position  is  changing.     When  the  molecule  arrives  at 

the  position  2,  Xh^ potential  energy  represented  by  /   Idp  has  been 

converted    into   energy  of  motion,  called   kinetic  energy,  which   is 

measured  by     ^  ^' ■  ,  the  kinetic  energy  of  the  molecule 

being     ^  ^'   at  i  and     ^  "^   at  2.    ' 

2  2 

HQnce,  potential  energy  is  defined  to  be  that  part  of  the  energy 
of  a  system  which  it  possesses  by  virtue  of  the  relative  positions 
of  its  different  masses,  and  kinetic  energy  to  be  the  energy  which 
the  system  possesses  by  virtue  of  the  motions  of  its  different 


64  MECHANICS   OF  SOLIDS. 

masses.     The  term  work  is  applied  to  the  change  of  energy  from 
one  form  or  body  to  another. 

For  example,  let  the  system  be  composed  of  a  unit  of  7nass 
and  the  earth,  and  let  the  limits  be  determined  by  two  horizontal 
planes  separated  by  a  distance  of  lo  feet.  Taking  the  body  at 
the  upper  limit,  we  have 


/> 


32.2  X  10  =  322  ft.-lbs. 


If  the  body  start  from  rest  and  fall  freely  in  vacuo  to  the  lower 
plane,  we  shall  have  this  potential  energy  converted  into  kinetic 
energy,  or  ^ 


mv        V 

—  =-  =  322: 
2  2 


r 


while  at  any  intermediate  point,  part  of  the  energy  is  potential 
and  part  kinetic.  Thus,  when  the  body  has  fallen  to  a  point 
midway  between  the  two  limits,  its  potential  energy  with  respect 
to  the  lower  plane  is  161  tt.-lbs.,  and  its  kinetic  energy  is  also 
161  ft.-lbs.  Each  form  of  energy  is  measured  in  units  of  work, 
but  no  work  is  done  unless  there  be  a  transformation  of  energy.  This 
illustrates  what  is  meant  h^  energy  of  position  and  energy  of  motion. 

To  illustrate  further,  the  muscular  potential  energy  in  a  man's- 
arm  may  be  changed  into  potential  energy  of  elasticity  in  a  bent 
bow,  and  the  potential  energy  of  the  bow  may  be  changed  into- 
kinetic  energy  of  a  moving  arrow,  work  being  done  in  both  cases. 
Kinetic  energy  cannot,  however,  be  transferred  from  one  body 
to  another  without  passing  through  the  potential  form. 

67.  The  Law  of  the  Conservation  of  Energy. — Scientific  investi- 
'gation  points  to  the  conclusion  that  the  total  quantity  of  energy 
in  the  universe,  as  well  as  the  total  quantity  of  matter,  is  invari- 
able; that  is,  that  neither  matter  nor  energy  can  be  created  or 
destroyed  by  any  known  means.  Accepting  this  as  a  scientific 
truth,  we   must  admit  that    the    energy  gained   or  lost  in  any 


WORK  AND  ENERGY.  65 

limited  system  of  masses  in  which  the  energy  varies  must  have 
been  obtained  from  other  masses  or  transferred  to  them. 

A  conservative  system  is  one  containing  a  certain  definite 
amount  of  energy.  It.  consists  of  limited  masses  subjected  to 
the  action  of  definite  forces.  The  law  of  energy  for  such  a  sys- 
tem is 

n-^rK=C, (93) 

in  which  TL  represents  the  potential  and  K  the  kinetic  energy  at 
any  time,  and  C  the  constant  quantity  of  energy  in  the  system. 
II  and  K  may  both  vary  with  the  time,  but  C  is  constant  ;  and  if 
any  change  occur  in  the  potential  energy  we  shall  have  a  cor- 
responding and  equal  but  opposite  change  in  the  kinetic  energy; 
thus, 

Ut.  ~-Ut»  =  Kt>f-Kt', (94) 

each  member  representing  the  change  in  the  corresponding  en- 
ergy during  the  time  /"  —  /'.  During  this  time  the  forces  of  the 
system  act  upon  the  masses  and  cause  them  to  change  their  con- 
ditions of  motion  and  relative  positions,  the  change  in  the  po- 
tential energy  being 

'fj'P (95) 


:2 


The  change  in  the  kinetic  energy  of  a  single  molecule  of  the 
system  during  this  interval  is 

£,'4?'^'' (96) 

and  for  the  whole  system, 

^S!..'"%^'- (97) 

The  change  in  the  potential  energy  in  the  time  dt  is  evi- 
dently 

:2idp (98) 

5 


66 


MECHANICS   OF  SOLIDS. 


and  the  change  in  the  kinetic  energy  during  the  same  time  is 

-^^^^^•^ (99) 

Since  these  two  quantities  are  always  equal,  we  have 

:EIdp  =  :2m^^,ds; (E) 

an  equation  which  expresses  the  Law  of  the  Conservation  of  Energy. 
This  law  may  be  stated  as  follows: 

The  total  energy  of  any  conservative  system  is  a  quantity  which  can- 
not be  increased  or  diminished  by  any  mutual  action  of  the  bodies  of  the 
system,  and  any  change  of  either  potential  or  kinetic  energy  must  always 
be  accompanied  by  an  equal  change  in  the  other* 

It  is  evident  from  this  statement  that  the  universe  is  the  only 
rigidly  conservative  system.  But  many  limited  systems  are  so 
remote  from  all  other  bodies  that  the  effect  of  these  latter*  is 
insignificant  when  considering  the  relative  motions  of  the  former. 

Eq.  (E)  is  the  fundamental  equation  of  mechanics,  and  it 
involves  all  relative  changes  in  the  configuration  and  motion  of 
any  conservative  system. 

68.   The  Principle  of  Virtual  Velocities. — If  no  change  of  state 

occur  in  any  of  the  molecules,  the  factors  —^  will  each  become 
zero,  and  the  equation  reduces  to 

:SIdp  =  o',  (S) 

or,  the  total  quantity  of  work  done  by  the  forces  upon  the  system 
of  masses  is  zero.  Any  one  of  the  elementary  quantities  of 
work  represented  by  the  type-symbol  Idp  is  exactly  equal  in 
amount,  but  of  a  contrary  sign,  to  the  aggregate  quantity  of 
work  of  all  the  other  forces  represented  by  ^/V/.  Such  a 
system  of  forces  is  said  to  be  in  equilibrio,  and  the  masses  in 
equilibrium.     If  the  latter  be  in  motion,  this  motion  must  be 


WORK  AND  ENERGY.  67 

uniform.     Regarding  the  intensities  of  forces  as  always  positive, 

the  sign  of  the  products  Idp  depends  on  the 

sign   of  dp.     The  sign   of  dp  is   taken  positive  fny-^^-^m' 

when  it  falls  on  the  action-line  of  the  force,      « — ^ — ' E » 

and   negative  when   it  falls   on   the  action-line 
produced  (Fig.  39). 

The  elementary  paths  whose  projections  are  dp  are  called 
virtual  velocities.  Being  the  actual  paths  described  in  the  time  dt, 
they  have  the  same  ratio  to  each  other  as  the  velocities  of  the 
points  of  application  at  the  instant  considered. 

The  products  Idp^  J'dp\  etc.,  are  called  virtual  moments;  they 
are  the  elementary  quantities  of  work  done  by  the  forces  while 
their  points  of  application  move  over  the  distances  whose  pro- 
jections on  the  action-lines  are  dp^  dp\  etc. 

Equation  (S)  is  the  form  taken  by  the  fundamental  equation 
in  Statics,  and  is  the  mathematical  statement  of  the  principle  of 
virtual  velocities;  that  is,  when  any  system  of  forces  is  in  equilibrio 
the  algebraic  sum  of  their  virtual  moments  is  equal  to  zero.  In  such 
a  system  the  potential  energy  is  constant,  none  being  trans- 
formed into  kinetic  energy. 

The  converse  of  this  principle  is  also  true;  that  is,  when  the 
algebraic  sum  of  the  virtual  moments  of  any  system  of  forces  is 
equal  to  zero  the  forces  are  in  equilibrio. 

69.  Equation  (E)  referred  to  Rectangular  Co-ordinate  Axes. — 
Let  a,  b,  c  be  the  angles  made  by  the  virtual  velocity  of  the 
point  of  application  of  a  force  with  the  axes,  and  d  the  angle 
between  this  virtual  velocity  and  the  action-line  of  the  force. 
Then  we  have 

cos  d  =  cos  a  cos  a  -j-  cos  P  cos  b  -{-  cos  y  cos  c\      ,     (100) 

and  multiplying  by  ds^ 

ds  cos  d  =  cos  ads  cos  a  -\-  cos  fids  cos  b  ■\-  coc  y^^  cos  ^,(101) 

or 

dp  =  cos  adx  -\-  cos  /3dy  +  cos  ydz (102) 


68  MECHANICS  OF  SOLIDS. 

Multiplying  both  members  of  this  equation  by  the  intensity 
of  the  force,  we  have 

Idp  =  /  cos  adx  +  /  cos  ^dy  -\- 1  cos  ydz.    .     .     (103) 

That  is,  f/ie  virtual  moment  of  any  force  is  equal  to  the  sum  of  the 
virtual  moments  of  its  rectangular  components. 
For  the  whole  system  we  have 

'^Idp  =  ^/  cos  adx  +  -2"/  cos  ^dy  +  ^I  cos  ydz.      (104) 

70.  Let  «',  b'^  c'  be  the  angles  made  by  the  elementary  path 
of  any  molecule  with  the  axes,  and  we  have 

I  =  cos'^  «'  +  cos'^  b'  +  cos''  c'y  .     ,     ,    ,     (105) 
and,  multiplying  by  m—=-ds^ 

d^s .  d'^s  ,         a     -    ,      d^s .         »  ,,   ,       d^s  J         »    .       ,     .. 

^n-zpds  =  ^;^^-f  cos"  a'  +  ^-rpf^^  cos'  h'  +  ^-j^ds  cos'  r .     (106) 

But 

ds  cos  a'  =  dx;        ds  cos  b'  =  dy;        ds  cos  c'  =  dz\    )  /  ^\ 
^V  cos  a*  -  d'x\     d^s  cos  b'  =  d''y\    d'^s  cos  .;'  =  d'^z.  )  ^     ^' 

Hence 

d'^s  ,  d^x  .     .       d^y ,     ,      d^z.  ,     .. 

That  is,  the  increment  of  the  kinetic  energy  of  any  molecule  is  equal 
to  the  sum  of  the  increments  estitnated  in  any  three  rectangular  direc- 
tions. 


WORK  AND  ENERGY. 


69 


For  the  whole  system  we  have 


'2m— ^ds  =  2m—pdx  +  2m-^^dy  +  2m— ^dz.    .     (109) 


Substituting  in  Eq.  (E),  we  have 


21  cos  adx  4"  -2"/  cos  /3dy  -\-  21  cos  ydz  = 

~dF        +       ^p"^     '^^^^^dP    '    ' 


(110) 


(^. 


/ 


x^ 


"7\ 


This  transformation  has  not  in  any  way  affected  the  gener- 
ality of  Eq.  (E)  which,  in  its  new  form,  still  embodies  all  the  cir- 
cumstances of  motion  of  the  molecules  of  a  body,  or  of  a  sys- 
tem of  bodies,  under   the  action  of  any  system  of  extraneous 
forces  whatever. 
O^'    71.  Application  of  Equation  E  to  the  Motion  of  a  Rigid  Solid. — 
A  rigid  solid  is  a  body  whose  molecules  are  supposed  to  preserve 
unchanged  their  relative  distances  from  each   other.     The  most 
general  motion  that  can  be  im- 
agined .for  such  a  hypothetical  2  z' 
solid   is  one  compounded  of   a 
motion  of  translation  and  rota- 
tion.     Its  motion  of  translation                      ,/       I 
may  be  defined  by  that  of  one 
of  its  molecules,  and  its  motion 
of  rotation  by  that  of  the  body 
about    this    molecule.     In    Fig. 
40  let  O  be  any  fixed  origin,  O' 
the    position   at    any   instant   of                            Fig.  40. 
the    particular    molecule    which 

determines  the  motion  of  translation,  and  ;//  the  position  of  any 
other  molecule  at  the  same  instant.  Let  ic,  y,  z  be  the  co- 
ordinates of  ;;/  referred  to  the  fixed  origin;  x^^y^,  z^,  the  co-ordi- 
nates of  the  movable  origin  6>' referred  to  the  fixed,  and  x\y',  z' 
the  co-ordinates  of  m  referred  to  the  movable  origin. 


X' 


4/ 


70 


MECHANICS  OF  SOLIDS. 


Then,  supposing  the  axes  at  the  movable  origin  O'  to  be 
always  parallel  to  the  fixed  axes  at  6>,  we  have 

x  =  x^^rx'\       y=yo+y';       ^  =  -^0  +  ^';     •    (m) 

dx  =  dx^  +  dx';     dy  =  dy^-\-  dy*\     dz  —  dz^  +  dz\  .     (112) 

Measuring  the  angles  about  O*  as  indicated  in  the  figure  to 
conform  to  Art.  35,  we  have,  for  the  increments  of  x\y\  z\  due 
to  rotation  about  the  axes  X\  K',  Z', 


dd 


X\ 


dd=o\ 


^dS  =  -  add  sin  ^  =  -  z'dd-, 
du 


dO  =      add  cos  d  =      fdO. 


djl 
dd 

dx' 

—  dtp=.       cdi/)  cos  tp  =       z^dtp ; 

dz' 

—  dtp=—  cdip  sin  <p  =  —  x'dtp. 


f  dx^ 


d<p=  —  bd(p  sin  0  =  —  y^0; 
d(f}=.       ddtpcos(f>=       x'dcp; 


dcf) 
d^ 


("3) 


(114) 


("5) 


Hence  we  have,  for  the  total  differentials, 


dx'  =  z'd^-/d(p', 
d/  =  x'dtp-  z'dd; 
dz'  =yde  -  x'dip. 


(116) 


WORK  AND  ENERGY. 


71 


Substituting  these  values  in  Eqs.  (112),  we  have 


dx  =  dx^  +  z'd^  —y'd<p\ 
dy  =  dy^  J^x'd(p-  z'dd) 
dz=zdz^  -{-/dS  -x'dtp. 


(117) 


Since  these  values  apply  to  any  point  of  the  body,  we  may 
substitute  them  in  Eq.  (no),  and  we  have 


21  cos  a(dx^  -f-  z^dtp  —  y'd(f>) 
+  21  cos  P(dy^  +  x'd(t)  -  z'dd) 
+  21  cos  y{dz^  -\-yde  -  x'dip) 


^2m—-,(dx,^z'd^--yd<t>) 
J^2f/-^ldy,^x^d<t>^z'de) 
+  ^tn^^Xdz,  -VydB  -  x^dtp). 


(118) 


But  dx^^  dy^^  dz^  relate  to  the  movable  origin,  and  dd,  dtpy  d(j> 
are  independent  of  the  position  of  m  because  the  body  is  rigid  ; 
each  of  these  differentials  is  therefore  a  common  factor  of  the 
terms  which  it  enters,  and  Eq.  (118)  may  be  written 

{21  cos  a  -  ^ffi^dx,  +  {21  cos  /?  -  ^^5^)^A 

+  [21  cos  y  -  2m-^^dz, 

+  [^/(^'  cos  /?  -/  cos  a)  -  2m'^^^^^^^yil>  \  (i  19) 

I    V^Tf  f                    f           \        ^    z^d^x  —  x'd'z'l  ,, 
-\-     2il(^z'  cos  a  —  x'  cos  y)  —  2m -— \dip 

+  [:^/(/cos  y-z^  cos  /?)  -  2m^?-/'^'y']de  =  o. 


72 


MECHANICS   OF  SOLIDS. 


72.  Application  of  Eq.  (E)  to  a  Rigid  Solid ^  perfectly  free  to 
Move. — The  only  restriction  which  has  thus  far  been  imposed  is 
that  this  equation,  which  has  been  derived  from  Eq.  (E),  shall 
apply  'to. a -single  rigid  solid  acted  upon  by  any  extraneous 
forces.  The  body  may  be  subjected  to  any  conditions  whatever 
as  to  its  possible  motion  under  the  action  of  these  forces,  and 
the  values  of  dx^,  dy^^  dz^,  dd,  dij;,  dcf)  will  depend  on  these  con- 
ditions. If  no  conditions  be  imposed,  that  is,  if  the  body  be 
free,  then  dx^,  dy^,  dz^,  dd,  dip,  dcf),  will  be  entirely  arbitrary  and 
independent  of  each  other.  Hence  we  have,  by  the  principle  of 
indeterminate  coefficients, 


X  =  21  cos  a  =:  29 


dy 
dt'' 


Z  =  2/  cos  y  =  2m 

x^dy  —y^d^x 


Y=  21  cos  /3  =  2m -f-,;    (- 

d\^ 
dt^'' 


2J(x'  cos  fi  —  y*  cos  a)  ■=•  2 
2I(z'  cos  a  —  x'  cos  y)  =  277i 


21{y'  cos  y  —  z'  cos  ft)  =  2 


df 
z'd'^x  —  x'  d'^z 

_  y'd'^z  —  z'dy 


dt'' 


(T) 


(R) 


73.  Interpretation  of  Equations  (T)  and  (R). — These  six  condi- 
tions, applicable  to  the  case  of  a  free  rigid  solid,  having  been 
derived  from  the  general  equation  of  energy,  embody  all  the 
circumstances  of  motion  of  its  molecular  masses,  caused  by  the 
action  of  extraneous  forces.  Considering  Eqs.  (T),  we  see  that 
their  rniddle  members  are  the  sums  of  the  component  intensities 
of  the  extraneous  forces  in  the  directions  of  the  rectangular  axes, 
and  the  last  members  are  the  sums  of  the  products  of  the 
molecular  masses  of  the  body  by  their  accelerations  in  the  cor- 
responding directions.     But  these  products  are  the  type-symbols 


WORK  AND  ENERGY.  73 

of  the  intensities  of  extraneous  forces  acting  on  the  molecular 
masses.  Hence  in  a  free  rigid  solid  any  system  of  extraneous 
forces  may  be  replaced  by  an  equal  set  whose  points  of  applica- 
tion are  the  molecules  of  the  body,  and  the  circumstances  of  the  >. 
motion  of  translation  of  the  latter  will  not  be  changed.  We  see 
also  that  whether  there  be  one  or  many  extraneous  forces  acting 
on  the  solid,  the  connections  which  unite  its  molecules  together 
cause  the  effect  of  these  forces  to  be  distributed  throughout  the 
whole  body.  Eqs.  (T)  therefore  refer  to  motion  of  translation,  and 
express  the  fact  that  the  algebraic  sum  of  the  component  intensities  of 
the  extraneous  forces,  estimated  in  any  direction,  is  measured  by  the  sum 
of  the  products  of  the  mass  of  each  molecule  by  its  acceleration  in  that 
direction. 

Referring  now  to  the  second  members  of  (R),  and  consider- 
ing the  first  of  these,  we  see  that  it  is  the  summation  of  terms  of 
the  form  of 

,  .d'^y  ,  ,d*x  .       . 

But  ^-^'—f-i  is  the  measure  of  the  intensity  of  the  force  which  for 
the  instant  dt  acts  on  m'  in  the  direction  of  the  axisjj'',  and  x*  be- 

7  2 

ing  the  co-ordinate  of  ;;/'  referred  to  O' ,  the  product  m'x*      ,   is 

the  moment  of  that  force  with  respect  to  the  axis  z* .     Similarly 

d'^x    .      ,  .     .       .  ,d^x 

-j-j-  IS  the  moment  of  the   force  ^^t—-^ 


d'^x  d^x 

m'y'-j-^  is  the  moment  of  the   force  w'— jy-  with  respect  to  the 


force  acting  on  ;//  at  that  instant  with  respect  to  the  axis  z\ 
We  see,  therefore,  that  each  molecule  may  be  regarded  as  being 
subjected  to  a  force  of  certain  intensity,  and  the  algebraic  sum  of 
the  moments  of  these  forces  at  any  ifistant,  with  respect  to  any  axis,  is 
exactly  equal  to  that  of  the  extraneous  forces  at  the  same  instant  with 
respect  to  the  same  axis  ;  and  this  is  what  is  expressed  by  Eqs.  (R). 
74.  If  the  solid  be  not  free,  tlie  conditions  to  be  satisfied  are 
less  than  six  in  number.      For  example,  if  one  point  of   the  body 


74  MECHANICS   OF  SOLIDS. 

be  assumed  as  fixed,  this  point  may  be  taken  as  the  origin  0\ 
and  we  shall  have 

dx^  =  dy^  =  dz^  =  o. 

The  first  three  terms  of  Eq.  (119)  then  reduce  to  zero;  and 
since  dd^  dtp,  d(f)  are  still  arbitrary  and  independent,  this  equa- 
tion is  satisfied  when  the  three  conditions  (R)  are  satisfied. 

If  two  points  be  fixed,  the  right  line  joining  them  will  be  a 
fixed  axis,  and  may  be  taken  as  the  axis  Y'.  We  shall  then  have 
dx^  =.  dy^=.  dz^  =  o,  and  d^  =  ^^  =  o;  and  since  dtp  is  not  neces- 
sarily zero,  Eq.  (119)  is  satisfied  when  the  second  of  Eqs.  (R)  is 
satisfied;  that  is,  the  single  condition  of  rotation  about  the  fixed 
axis  ;  and  similarly  for  other  conditions  of  constraint. 

75.  If  the  molecules  of  the  solid  be  in  uniform  motion  or  at 
rest,  the  forces  and  moments  are  balanced,  and  the  body  is  in 
equilibrium  both  as  to  translation  and  rotation.  Eqs.  (T)  and 
(R)  then  become 

X  ==■  2/  cos  £x  =  o;\ 

V=::S/cos/3  =  o;y (T') 

Z  =  2/  cos  y  =z  o;) 

Vx  —  Xy  =  2/{x^  cos  /3  —  y'  cos  or)  =  o;  ) 

Xz  —  Zx  =  '2l{z'  cos  a  —  x'  cos  ;/)  =  o;  )■    .     ,     (R') 

Zy  —  Yz  =  2/{y'  cos  y  —  z^  cos  /?)  =  o;  ) 

which  are  the  six  conditions  of  equilibrium. 

76.  Analytical  Mechanics  consists  essentially  in  the  application 
of  Equations  (E),  (T),  (R),  (T')  and  (R')  to  conservative  systems 
of  masses  and  forces.  The  object  of  the  discussion  is  to  ascer- 
tain the  position  of  any  and  all  molecules  at  any  time,  the  nature 
and  direction  of  their  motions,  and  the  configuration  of  the 
bodies  of  which  they  are  the  elements.  The  theory  of  the  in- 
vestigation is  simple,  but  the  practical  application  is  limited  by 
our  mathematical  knowledge  and  skill. 


work:  and  energy. 


75 


77.  Equations  (T)  and  (R)  referred  to  the  Centre  of  Mass  as  a 
Movable  Origin. — Since  there  are  as  many  terms  in  the  last  mem- 
bers of  Eqs.  (T)  and  (R)  as  there  are  molecules  in  the  body, 
these  equations  are  not  in  a  convenient  form  for  discussion.  By 
taking  the  movable  origin  at  the  centre  of  mass,  the  resulting 
equations  are  no  less  general  than  before,  while  the  solution  of 
practical  problems  is  much  simplified. 

78.  Equations  of  Translation. — In  the  first  of  Eqs.  (T), 


d^'x 
X  =  J^/  cos  a  =  ^^-77-% ' 


(12.) 


Substitute  for  d*x  its  value  obtained  by  differentiating  Eq.  (112), 
and  we  have 


d^x  d'^x  d'^x' 

X  =  J?/  cos  a  =  ^m-j^  =  2m-^  -f  .^;//— 5-.  .     (122) 


dt' 


dt' 


But  d*x^  is  a  common  factor  in  the  term  which  it  enters,  and 
from  the  principle  of  the  centre  of  mass,  Eqs.  (64),  we  have 


2md^x' z^  Md*x, (123) 


and  Eq.  (122)  reduces  to 


d^x  d^x 

X=:SIcos  a  =  m"^  +  M~. 
at  at 


("4) 


_    Taking  the  movable  origin  at  the  centre  of  mass,  we  have 
^  =  o,  and  Eqs.  (T)  become 


Ar=^/cosar  =  i^-'^'"^»   ^ 


dt^ 
Y=2I cos  fi  =  M-^ 


Z  =  ^I  cos  y  =  M- 


dt^ 

d\ 

dr 


(T.) 


76 


MECHANICS   OF  SOLIDS. 


from  which  the  motion  of  translation  of  the  centre  of  mass  of  a 
free  rigid  solid,  under  the  action  of  incessant  forces,  can  be  found. 
If  the  forces  be  impulsions^  then  Eqs.  (T)  become         .    , 


X  =  2/^  cos  a  =  ^ni—-' 

y=  :2I^  cos  /3  =  2m^-; 

'    Z  =  ^I.  cos  y  =  ^m-r-\ 
at 


which,  when  the  movable  origin  is  the  centre  of  mass,  become 


dx 
X=  21,  COS  a  =  AI-^  =  MFy, 

Y=  21,  COS  /?  =  m"^  =  MVy\ 

Z  =  2/,  cos  r  =  M^  =  MV,  ; 


dt 


(T.') 


from  which  the  motion  of  translation  of  the  centre  of  mass  of  a 
free  rigid  solid,  under  the  action  of  impulsions,  can  be  found. 

From  Eqs.  (T^,)  and  (Tq/)  we  draw  the  following  conclu- 
sions: 

(i)  That  the  motion  of  the  centre  of  mass  of  a  free  rigid  solid., 
under  the  action  of  extraneoics  forces^  is  entirely  independent  of  the 
relative  positions  of  the  molecular  masses,  since  their  co-ordinates  have 
disappeared  from  the  equations  of  motion. 

(2)  That  the  motion  of  the  centre  of  mass  depe7ids  only  upon  the 
mass  of  the  body  and  the  intensities  and  directions  of  the  extraneous 
forces,  and  is  i7idependent  of  the  points  of  application  of  the  forces. 

(3)  That  the  motion  of  the  centre  of  mass  will  be  precisely  the  same 
as  that  of  a  material  point  whose  mass  is  equal  to  that  of  the  body.,  sub- 


WOJ^K  AND  ENERGY. 


77 


----■^R' 


jected  to  the  action  of  forces  equal  to  the  given  forces  in  intensity  and 
having  the  same  direction. 

79.  To  illustrate  tliese  conclusions,  let  us  consider  the  motion 
of  translation  of  the  centre  of  mass  of  the  free  rigid  solid  ^4,  Fig.  41, 
first,  under  the  action  of  the 
several  incessant  forces /^^jP", 
P'"  and  /*'",  and,  second,  under 
the  action  of  the  impulsions 
^/,  Pn^  ^iu  and  P,,. 

Let  O  be  the  centre  of 
mass,  and  let  forces  equal  to 
the  extraneous  forces  in  inten- 
sity and  direction  be  supposed 
applied  to  it.  Let  R'  be  the 
resultant  of  those  forces  which 
have  O  as  their  common  point  of  application;  then  from  the  above 
principles  it  can  be  asserted  that  under  the  action  of  the  given 
incessant  forces  the  centre  of  mass  will  move  along  the  right  line 

R* 

OR*  with  a  constant  acceleration  equal  to  -7>;  and  if  the  forces  be 

impulsions,  that  the  centre  of  mass  will  move  along  the  right 

R 

line  OR^  with  a  constant  velocity  equal  to  -~=. 

80.  Equations  of  Rotation. — When  the  forces  are  incessant  and 
the  movable  origin  is  the  centre  of  mass,  Eqs.  (R)  readily- 
red  uce  to 


Fig 


L-  Yx  -  Xy=:SI{x'  cos  /?-/ cos a)='Sf 
M=Xz  —  Zx=i^I{z'  cos  a—  x' cosy)— '2m 
iV=  Zy-Yz  =  ^/(/cos  y-  z'  cos  fi)  =  2m 


x'dy-yd'x'  ^ 


df 

z'd'x'-x'd'z' 

dt' 

yd\'-z'dy 

dt^ 


;  KR«) 


To  show  this,  let  us  reduce  the  second  member  of  the  first  of 
Eqs.  (R);  we  have 


78 


MECHANICS  OF  SOLIDS. 


^    x'd'y-fd'x      „    x'd'y.-Y  x'dy -  Vd'x- y'(fx' 

^"^      dt'       =  ^'" Te 


__  d'^y^^vix'  —  d'^x^'^my'  -}-  ^mx'd^y'  —  ^my'd^x' 

x'dW  -  y'd'^x' 
=  ^^ le ' 


since,  by  the  principle  of  the  centre  of  mass,  ^mx'  =  2my'  =  o. 
Similarly,  w/ieu  the  forces  are  wipulsions^  Eqs.  (R)  become     V 


Z,=:  Yx-Xy^  :2l,(x'  cos  y^-/  cos  a)  =  '^m'^^jf—'  ^ 

M^=Xz—Zx=:2lXz'  co^  a—  x'  cosy)  =  :2m ;  }{^2^) 

N,=Zy  -  Vz==2I^{y  cos  y  -  z'  cos/3)  =  2^/"^'  ~  ^'^-^ 


which,  when  referred  to  the  centre  of  mass  as  a  centre  of  rota- 
tion, become 


L-  Yx-Xy^2I,{x'  cos  /?-/  cos  a)  =  ^nT  ^-^    /  ^^  ; 


M^^^Xz  —Zx=2I^(z'  cos  a  —x'  cos  y)  =  2? 


N^=Zy  —  Yz=2/,{y' cos  y—z*  cos  /3)  =  2 


dt 

'dx'-x'dz' , 

d~t       ' 

_         /dz'-z'd/ 


MR-') 


dt 


8l.  If  one  point  of  the  body  be  fixed,  we  have,  by  taking  it 
as  the  origin  0\ 

^»^  =  ^V;    dy  =  dy;    d'zz=dV; 


and  Eqs.  (R)  reduce  to  the  form  of  Eqs.  (RJ)  or  (Rm')>  independ- 
ently of  the  principle  of  the  centre  of  mass. 

Also,  if  an  axis  be  fixed,  that  one  of  Eqs.  (R)  which  applies 


GENERAL  THEOREM  OF  ENERGY.  79 

to  the  axis  in  question  reduces  to  the  corresponding  one  of  Eqs. 
(Rjn)  or  (Rni')'  according  as  the  forces  are  incessant  or  impulsive. 

82.  In  Eqs.  (Rm)  and  (Rm')  the  co-ordinates  of  the  centre  of 
mass  do  not  appear;  therefore  the  motion  of  rotation  of  the  body 
about  the  centre  of  mass  is  independent  of  the  position  of  this 
centre,  and  will  be  the  same  whetlier  it  be  considered  at  rest  or 
in  the  state  of  its  actual  motion.  This  exhibits  the  complete 
independence  of  the  motions  of  translation  and  of  rotation,  and 
permits  the  investigation  of  either  as  if  the  other  did  not  exist. 
By  means  of  Eqs.  (Tm)  the  position,  velocity  and  acceleration  of 
the  centre  of  mass  can  be  theoretically  determined  at  any  time, 
and  by  Eqs.  (Rm)  the  corresponding  positions,  angular  velocities 
and  accelerations  of  every  molecule  with  respect  to  the  centre  of 
mass;  and  thus  the  configuration  of  the  w^hole  body  about  this 
point  can  be  determined  for  the  same  instant.  The  mathematical 
difficulties,  however,  due  to  integration,  limit  their  application  to 
but  a  few  simple  cases.  It  is  to  be  noted  that  when  the  problem 
involves  incessant  forces,  Eqs.  (T^)  and  (Rm),  and  when  impul- 
sions alone,  Eqs.  (T^')  and  (Rm')>  ^^^  ^^  ^^  used. 

General  Theorem  of  Energy  applied  to  a  Free  Rigid 
Body  whose  Centre  of  Mass  is  referred  to  a  Fixed 
Point  in  Space. 

83.  Translation  under  Incessafit  Forces. — Multiply  Eqs.  (T^)  by 
dx,  dy,  dz,  respectively,  add  the  results  and  integrate,  and  we 
have 

fiX^.  +  Ydy  +  Zdi)  =  j^,f^J^^:f+^y±J^ 

=  f(^|:±^)+C=^Vc,(.6) 

in  which  x^y,  z  are  the  co-ordinates  of  the  centre  of  mass  referred 
to  the  fixed  origin,  V  is  the  variable  velocity  with  respect  to  the 
sar.ie  point,  and  Af  is  the  mass  of  the  body.     The  first  member, 


80  MECHANICS   OF  SOLIDS. 

J (Xdx  ■\-  Ydy -\- Zdz), (127) 

is,  Art.  (i(i,  an  expression  for  the  quantity  of  work  done  by  the 
extraneous  forces,  or  the  total  quantity  of  potential  energy  ex- 
pended by  them,  and  whose  particular  value  in  any  case  will  be 
determined  when  the  limits  of  the  integration  are  fixed.  The 
constant  C  of  the  second  member  is  evidently  the  kinetic  energy 
which  existed  in  the  body  at  the  instant  the  extraneous  forces 
began  to  act;  for  at  that  instant,  say  /j,  their  work  is  zero,  and 
if  Fj  be  the  corresponding  velocity,  then 

C^-'JLLi. (i23) 

2  ^       ' 

The  first  term  of  the  second  member, ,  is  the  total  ki- 

2 

netic  energy  at  any  time,  and  hence  the  whole  second  member, 

—- -^ (129) 


being  the  difference  between  that  possessed  by  the  body  at  any 
time  and  that  when  the  extraneous  forces  began  to  act  on  the 
body,  is  the  exact  equivalent  of  the  potential  energy  represented 
by  the  first  member.  Here  we  have,  as  should  have  been  ex- 
pected, the  general  law  of  the  transformation  of  energy. 

84.  The  first  member  may  be  integrated  when  the  compo- 
nent forces  in  the  directions  of  the  co-ordinate  axes  are  constant, 
and  when,  if  these  forces  be  variable,  it  becomes  a  known  differ- 
ential function  of  the  three  co-ordinates  .r,  7,  z.  In  the  first  case, 
let  R  be  the  constant  intensity  of  the  resultant  of  the  forces,  and 
a,  b^  c  the  angles  which  its  action-line  makes  with  the  co-ordi- 
nate axes.     Then  we  have  between  the  limits  1  and  2 

R(x  cos  a  +/  cosb-\-z  cos  ^),''=  F'(x,y,  z)^=M^—^ ^—',  (130) 


GENERAL   THEOREM  OF  ENERGY.  8 1 

In  the  second  case,  in  order  that  integration  may  be  possible, 
the  intensities  of  the  forces  must  be  functions  of  x,y,  2,  and  we 
have 

r\x,y,z):  =  M^-^^:^ (131) 

Hence  we  may  write  Eq.  (126)  in  either  case 


F(.^,y,i);  =  F{x^.\)-F(xj.^:^  =  M^^^-^,     (132) 

as  the  general  law  of  energy  when  forces  act  on  a  free  rigid 
solid  to  give  it  motion  of  translation,  under  the  conditions  im- 
posed above.  From  this  equation  we  conclude  that  the  velocity 
generated  in  a  free  rigid  solid  by  constant  forces^  or  by  variable 
forces  whose  intensities  are  functions  of  the  co-ordinates  of  the  centre 
of  mass ^  varies  only  with  the  values  of  the  co-ordinates;  and  that, 
should  the  centre  of  mass  ever  return  to  the  same  position  in 
space,  its  velocity  will  be  the  same  as  before,  whether  the  path 
by  which  it  reaches  this  point  be  the  same  or  not. 
If  the  forces  be  in  equilibrio,  then,  Art.  68, 

Xdx^Ydy^Zdz:=.o, (133) 

and  we  have 

=  a  constant; (^34) 

that  is,  the  velocity  is  constant. 

85.  Rotation  under  Incessant  Forces. — If  we  multiply  Eqs.  (Rm) 
by  dcf),  dtp  and  dd^  respectively,  add  the  results  and  reduce  by 
Eqs.  (116),  we  obtain 


21  cos  adx'  +  21  cos  fidy'  -\-  21  cos  ydz' 

^    (dx'd\x'  +  dy'dy  +  dz'd^z' 


),  (135) 


82  MECHANICS   OF  SOLIDS. 

which,  by  integration,  becomes 

J:2Idp  =jRdr  =  i2m[^^^^±^±^)+C'=i2mv'+C',  (136) 

when,  Eq.  (104),  we  replace  the  sum  of  the  virtual  moments  of 
the  extraneous  forces,  2Idp,  by  the  virtual  moment  of  the  result- 
ant, J^dr,  But  dr,  being  the  elementary  path  of  the  point  of  ap- 
plication of  the  resultant  projected  on  the  action-line  of  the  re- 
sultant, is  equal  to  the  product  of  the  path  described  by  a  point  at 
a  unit's  distance  from  the  centre  of  mass  by  the  lever  arm  of  i?. 
Let  k  be  this  lever  arm,  and  ds  the  elementary  arc  at  a  unit's 
distance,  and  we  have 

fj^Ms  =  i2mv'  +  C (137) 

The  first  member  is  the  general  expression  for  the  work  of 
rotation  done  by  the  resultant  of  the  system,  or  the  potential 
energy  transformed;  ^^mv^  is  the  kinetic  energy  of  rotation  of 
the  body  with  respect  to  the  centre  of  mass;  C  is  the  kinetic 
energy  of  rotation  in  the  body  before  any  has  been  transferred 
to  it  by  the  extraneous  forces,  and  is  equal  to  —  \^mv^\  there- 
fore we  hflve' 

CRkds  =  ^^mv"  -  i'Smv*      ...     .     (138) 

for  the  total  kinetic  energy  of  rotation  put  into  the  body  by 
the  expenditure  of  the  equivalent  amount  of  potential  energy 

likds. 


s 


86.  Adding  together  Eqs.  (132)  and  (138),  we  have 
£\xdx  +  ydy  +  Zdz)  -^-fjjRkds 


=  M^ -^-  -f  i:Sm(v,'  -  v:),     (139) 


GENERAL  THEOREM  OF  ENERGY.  83 

in  which  the  first  member  expresses  the  total  expenditure  of 
potential  energy  of  the  extraneous  forces  in  producing  motion 
of  translation  and  of  rotation,  and  the  second  member  the  equiv- 
alent quantity  of  kinetic  energy  which  has  resulted  therefrom. 
If  the  action-line  of  the  resultant  of  the  extraneous  forces  pass 
through  the  centre  of  mass,  the  second  term  of  the  first  member 
and  the  second  term  of  the  second  member  become  zero. 

Thus  we  see  that  the  poi7it  of  application  is  of  importance  in 
determining  the  effect  of  a  force,  since  by  supposing  it  to  be 
changeable  the  resulting  kinetic  energy  of  rotation  imparted  to 
the  body  will  be  correspondingly  varied. 

87.  Translation  under  Impulsive  Forces. — Squaring  Eqs.  (T^') 
and  adding,  we  have 

X«  -f  F»  +  Z'  =  i?/  =  M\  VJ  -f  F/  +  F,')  =  M'  V\  (140) 

whence 

R,-MV, (141) 

Hence,  when  a  free  rigid  solid  has  been  subjected  only  to  a 
system  of  .impulsions,  its  centre  of  mass  will  move  with  a  con- 

stant  velocity,  -7^. 

88.  Motion  of  Translation. — In  the  discussion  of  the  motion  of 
translation  of  bodies  two  classes  of  problems  arise.  In  the  first, 
which  are  called  direct^  we  have  given  the  mass  of  the  body  and 
the  forces  acting  upon  it;  and  it  is  required  to  find  the  path  of 
the  centre  of  mass  and  all  the  circumstances  of  its  motion.  In 
those  of  the  second  class,  which  are  called  inverse  problems,  the 
path  of  the  centre  of  mass  is  given,  and  it  is  required  to  find 
the  forces  which  will  cause  the  body  to  follow  that  path. 

89.  The  Direct  Proble7n. — To  solve  the  direct  problem  we  sub- 
stitute in  Eqs.  (Tn,)  the  mass  of  the  body  and  the  component  in- 
tensities of  the  forces,  and  obtain 


84 


MECHANICS  OF  SOLIDS. 


d'x 

X 

d^y 

V 

d'z 

Z 

de  " 

~  M' 

df 

M' 

de 

~  M' 

•  (142) 


in  which  the  accelerations  are  constant  or  variable  according  as 
the  forces  are  constant  or  variable.     Integrating,  we  have 


(■43) 


The  arbitrary  constants  in  these  equations  are  the  values  of 
the  component  velocities  when  /  =  o;  that  is,  at  the  epoch  or  in- 
stant from  which  /  is  estimated. 

Integrating  again,  we  have 


(144) 


D,  D*  and  D*'  being  the  co-ordinates  of  the  centre  of  mass  at 
the  epoch. 

The  values  of  C,  C,  C",  Z>,  D*  and  D"  are  called  the  initial 
conditions^  since  they  are  the  component  velocities  and  the  co- 
ordinates of  the  centre  of  mass,  when  the  forces  began  their 
action. 

The  integrals  in  Eqs.  (144)  can  readily  be  found  when  the 
^iven  forces  X^  Y,  Z  are  constants,  or,  if  variable,  when  they  can 
be  expressed  in  such  terms  of  /  as  to  make  them  known  integrable 


GENERAL  THEOREM  OF  ENERGY.  8$ 

expressions.  Then  by  eliminating  t  from  Eqs.  (144)  we  obtain 
two  equations  containing  x^  y,  z,  and  constants,  which  are  the 
equations  of  the  path.  The  problem  is  then  completely  solved, 
since  we  have  the  position,  velocity  and  acceleration  at  any  time, 
and  the  entire  path  of  the  body. 

90.  The  Inverse  Problem. — Since  the  centre  of  mass  may  de- 
scribe the  same  path  under  different  conditions  of  velocity  and 
acceleration,  the  inverse  problem  is  indeterminate.  It  may,  how- 
ever, be  made  determinate  by  assuming  the  initial  conditions 
and  one  component  velocity  or  one  component  acceleration. 

Let  the  equations  of  the  path  be 

By  differentiating  and  dividing,  by  dt,  we  obtain  two  equa- 
tions involving  three  component  velocities,  and  by  a  second  dif- 
ferentiation and  division  we  get  two  equations  containing  three 
component  accelerations.  To  obtain  an  equation  connecting 
the  velocity  with  the  component  accelerations,  differentiate  Eq. 
(126)  and  divide  by  the  differential  of  one  of  the  variables,  as 
dx.     We  thus  have 

or 

1  d{V')  _d*x       dy  dy       d'z  dz 

2  dx     ~  dt'''^  dt^'d'x'^  df^'dx    '     '     '     ^^"^^^ 

Now  if  one  of  the  component  velocities  be  assumed,  we  may 
obtain  the  value  of  Ffrom  the  equations  obtained  by  the  first 
differentiation  of  the  equations  of  the  path,  since 


---(ir+dr +(§)■• 


86  MECHANICS  OF  SOLIDS. 

Then  we  have  three  equations  in  which  the  three  component 
accelerations  are  the  only  unknown  quantities,  and  the  problem 
may  be  completely  solved. 

If  one  component  acceleration  be  assumed,  we  may  find  V 
from  Eq.  (147)  by  integration,  and  this  value,  together  with  the 
equations  involving  the  component  velocities,  makes  the  solu- 
tion possible.  Thus,  in  either  case  the  problem  is  determinate, 
and  all  the  circumstances  of  the  motion  may  be  found  as  in  the 
direct  problem. 

It  is  also  evident  that  the  problem  may  be  solved  by  assum- 
ing any  new  condition  connecting  the  six  unknown  quantities, 
since  we  already  have  five  equations  containing  them. 

91,  Examples  of  the  Direct  Problem. 

(i)  Constant  Forces. — Integrating  Eqs.  (142)  twice,  we  have 

dt-'M^^^'    Tt-M^^^^    di-^M^+^  ^     ('48) 


(149) 


Let  us  suppose  that  the  centre  of  mass  at  the  epoch  is  at  rest 
at  the  origin  of  co-ordinates;  then  Eqs.  (148)  and  (149)  become 

•  •     •     (ISO) 

•  .     .     (iSi) 
Eliminating  from  either  pair  of  equations  pertaining  to  the 


dx       X 
dt  ~  ~M  ' 

^y  -  y ,. 

dt  ~  M  ' 

dz       z 
dt  ~~  M  ' 

X  t' 

^"-J/2' 

^~ M2  ' 

Zt"" 

^~  M2' 

GENERAL    THEOREM  OF  ENERGY,  8/ 

same  axis,  as  x  for  example,  the  factor  -7^,  and  indicating  the  ve- 
locity in  that  direction  by  the  subscript,  we  have 


(152) 


Laws  of  Constant  Forces, — Eqs.  (150),  (151)  and  (152)  express 
the  laws  of  constant  forces,  which  are: 

I  St.  The  velocity  of  the  centre  of  mass  in  any  direction  varies 
directly  with  the  time,  and  at  any  instant  is  equal  to  the  product  of  the 
acceleration  in  that  direction  multiplied  by  the  time. 

2d.  The  space  passed  over  in  any  direction  varies  directly  as  the 
square  of  the  time,  and  at  any  instant  is  equal  to  the  acceleration  in  that 
direction  multiplied  by  half  the  square  of  the  time. 

3d.  The  space  described  in  any  direction  is  equal  to  the  component 
velocity  in  that  direction  at  the  time  considered,  multiplied  by  half  the 
time  since  the  epoch  ;  hence  the  space  described  in  the  first  unit  of  time  is 
equal  to  half  the  acceleration. 

92.  (2)  Motion  due  to  Gravity. — Let  the  weight  of  the  body, 
wliich  is  the  only  force  acting,  be  supposed  constant.  Take  the 
axis  of  z  vertical  and  positive  downward,  and  let  a,  fi,  y  be  the 
angles  which  the  weight  Mg  makes  with  the  axes  x^y^  2,  respec- 
tively.    Then  Eqs.  (148)  become 


d'^x      MfT  cos  a 

d\      Mg  cos  B  _ 

d^z       Mg  cos  y 


('53) 


Under  the  general  supposition  that  the  centre  of  mass  at  the 
epoch  is  in  motion  and  not  at  the  origin  of  co-ordinates,  these 
equations,  by  integration,  give 


88  MECHANICS  OF  SOLIDS. 

^=^    1=-''     %=^'^C',    .     .     .     (.54) 
and 

x=^Ct-\-D,    y=:Ct^D\     z:=^\gt'^C^t^D*\   (155) 

Hence,  Eqs.  (153),  the  accelerations  in  the  directions  of  x  and 
y  are  zero,  and  if  there  be  any  motion  in  a  horizontal  direction 
it  must  be  uniform;  this  is  also  shown  by  the  first  two  of  Eqs. 
(154).  The  acceleration  g  in  the  vertical  direction  is  that  due  to 
gravity;  and  the  velocity  in  this  direction  must  increase  alge- 
braically, as  shown  by  the  last  of  Eqs.  (154).  From  Eqs.  (155) 
we  see  that  the  distances  passed  over  in  the  directions  of  x  and  j 
vary  directly  with  the  time;  also  that  the  distance  of  the  centre 
of  mass  from  the  origin,  estimated  in  the  vertical  direction,  is 
composed  of  three  parts,  viz.,  the  initial  co-ordinate  D" ,  the 
space  due  to  the  initial  velocity  C",  and  that  due  to  the  constant 
effect  of  gravity. 

Confining  the  discussion  to  motion  in  a  vertical  direction, 
and  omitting  the  accents  from  the  constants,  the  equations 
become 


Jf=^;    Tt=St^C;    z  =  ^r  +  Ct  +  D.     .     (156) 


If  the  body  start  from  rest  at  the  origin,  C  and  Z>  will  both 
be  zero;  and  letting  v  represent  the  velocity  and  /i  the  height 
fallen  through,  we  have 

v  =  gt;    h  =  igf (157) 

Eliminating  /,  we  have 

V*  =  2gh (158) 


GENERAL    THEOREM  OF  ENERGY.  89 

This  relation  between  v  and  h  is  of  frequent  use  in  problems  of 
motion;  h  is  called  the  height  due  to  the  velocity  Vy  and  v  the  velocity 
due  to  the  height  h\  either  may  be  found  in  terms  of  the  other 
when  g  is  known. 

This  relation  may  also  be  obtained  from  the  general  law  of 
energy,  since  we  have 

wh  =  mgh  = , (159) 

or 

v"  =  2gh.  (160) 

If  the  body  be  projected  vertically  upward  from  the  origin 
with  an  initial  velocity  C,  then,  to  find  the  duration  of  its  ascent 
and  the  height  to  which  it  will  rise,  we  have,  Eqs.  (156), 

^     o=gt—Cy     or    /  =  -, (161) 


and 


h  =  igt'-Ct=-— (162) 


Gravity  therefore  abstracts  g  units  from  the  initial  velocity 
every  second  until  the  body  comes  to  rest  at  the  altitude  //,  after 
which  it  will  restore  g  units  of  velocity  each  second,  and  the 
body  will  reach  the  origin  with  its  initial  velocity. 

From  Eq.  (162),  as  also  from  Eq.  (158),  we  see  that  the  height 
which  the  body  will  attain  is  equal  to  tlie  square  of  the  initial 
velocity  divided  by  twice  the  acceleration  due  to  gravity. 

93-  (3)  ^^^  Trajectory  in  Vacuo. — The  path  described  by  the 
centre  of  mass  of  a  body  is  called  the  trajectory;  if  the  body  be 
given  an  initial  velocity,  it  is  called  a  projectile;  but  the  term 
trajectory  is  usually  limited  to  the  paths  of  projectiles  intended 
to  be  thrown  from  guns  by  means  of  some  explosive — generally 
gunpowder. 


90  MECHANICS  OF  SOLIDS. 

The  discussion  of  the  trajectory  in  vacuo  limits  the  forces 
acting  to  the  weights  alone;  and  since  for  short  distances  on  its 
surface  the  radii  of  the  earth  are  sensibly  parallel,  the  weight  at 
all  points  of  the  trajectory  may  be  considered  as  acting  parallel 
to  its  direction  at  the  origin.  Let  the  projectile  start  from  the 
origin,  and  take  the  axis  z  vertical  and  positive  upward;  the 
trajectory  will  lie  in  the  plane  of  this  axis  and  the  initial  direc- 
tion of  motion,  since  there  is  no  force  acting  oblique  to  this 
plane.  If  we  take  the  axis  of  x  in  this  plane,  the  differential 
equations  of  motion  become 

d^z        Z        Mg  cos  y  ,  ,  ^ 

d^x        X       M^  cos  a  ,  ^  ^ 

ir  =  M=        M       ^SCosa  =  o;    .    .    .    (164) 


since  y  =  i8o°  and  a  =  90°. 

These  equations  give,  by  two  integrations, 


X 


=  a;     2  =  -  igt' -i- C/.     ......     (166) 


Let  V  be  the  initial  velocity,  and  6  the  angle  which  the  tra- 
jectory at  the  origin  makes  with  the  axis  at  x.     Then  we  have 

C=  Fcos  6;     C"=  Ksin  6. 

Substituting  these  values  of  Eqs.  (165)  and  (166),  we  have 

^-=^Vcos6',      g=-^/+Fsin^;    .     .     .     (167) 
x=Fcosef;      0=  ~i^/»+  Tsin  ^/.    .     .     (168) 

Eliminating  /  from  Eqs.  (168),  we  find  the  equation  of  the 


GENERAL    THEOREM  OF  ENERGY.  9 1 

trajectory  to  be 

«=*""^-*?F^^i^ (169) 


or 


x" 


ZZ^X\.2XiQ 7 j-^, (170) 

4^  COS  Q  \  I  / 

when  for  V^  we  substitute  2gk. 

To  find  the  co-ordinates  of  the  highest  point,  we  have 

dz  ^  2X  ,        . 

~  =  o  —  tan  6 7 ^~2j,     ....     (171) 

dx  4/1  cos  6  \  I  / 

or 

X  =.  2k  tan  6  cos'  ^  =  -^  sin  2^.  .     .     .     .     (172) 

This  value  substituted  in  Eq.  {170)  gives 

,     .        n  /I       ^'  sin"  20 

z  •=.  h  sin  2u  tan  u ; —-p, 

4/1  cos    u 

=  2/4  sin'  e-/i  sin'  6  = /i  sin'  6.  ,     .     .     (173) 

Transfer  the  origin  to  the  highest   point,  without  changing 
the  directions  of  the  axes,  and  we  have  (H 1 1 

,    7    •   9  n       /      ,    r    .       /i\  /I       {x  -\-  h  sin  26^)'   ,       . 

2  4-  >4  sin'  ^  =  (a:  +  y^  sin  26)  tan  ^  —  ^— ^ 7-5-^,  (174) 

^  ^  4^  cos'  6      '  ^  '  ^/ 

or 

4^  cos'  ^js:+  4^'  sin'  ^  cos'  6^  =  4^  cos'  ^  tan  ^j; 

-j-4^'cos'  ^sin  2^  tan  ^—^'—2^ sin  2^;t— ^'sin*  2O,  (175) 

But 

4^'  sin'  6  cos'  (9  =  4>4'  cos'  d  sin  2(9  tan  6  ^  h*  sin'  2^,  (176) 


92  MECHANICS   OF  SOLIDS. 

and 

4^  cos'  d  tan  6  •=.  ih  sin  2^, {i77) 

and  Eq.  (175)  reduces  to 

x^  —  ^  ^h  cos''  Qz^    .     .     .    .     .     .     (178) 

the  equation  of  a  parabola  whose  axis  is  the  vertical  through  the 
highest  point. 

The  range,  which  is  that  portion  of  the  original  axis  of  x 
between  the  two  branches  of  the  curve,  is  seen  from  Eq.  (172)  to 
be  2/1  sin  26,  and  its  maximum  value  for  any  given  value  of  V 
is  obtained  when  the  angle  of  projection  is  45°;  and  since  this 
value  is  2h,  the  maximum  range  is  equal  to  twice  the  height  due 
to  the  initial  velocity.  The  corresponding  value  of  z  is  ^//,  or  |- 
of  the  maximum  range. 

The  time  required  to  describe  any  portion  of  the  curve  is  evi- 
dently 


The  time  from  the  origin  of  motion  to  the  highest  point  is 
also  given  by  the  second  of  Eqs.  (167), 

~=o^-gt+Vs\ne, (180) 

or 

t=^^^^. (.80 

g 

The  value  for  the  velocity  at  any  point,  obtained  from  Eqs. 
(167),  is 

Z;«  =  ^^'  +  ^<=:   F'+/^'-2F^/sin^,    .      .      (182) 


GENERAL    THEOREM  OF  ENERGY.  93 


or,  eliminating  thy  means  of  Eq.  (179), 


V  =  ^  V^  -  2g  tan  ex  +  j,ff^^.  ff     .     .     .     (183) 

From  the  symmetry  of  the  curve  with  respect  to  the  vertical 
through  the  highest  point,  it  is  evident  that 

(i)  The  two  branches  are  described  in  equal  times. 

(2)  For  points  at  the  same  height  the  angle  of  fall  is  the  sup- 
plement of  the  angle  of  rise. 

(3)  For  points  at  tlie  same  height  the  velocities  are  equal, 
since  the  horizontal  velocity  is  constant  and  the  vertical  veloci- 
ties are  numerically  equal  for  equal  values  of  z. 

In  Eq.  (170)  substitute  for  cos  6  its  value  in  terms  of  tan  6, 

and  we  have 

„        ^'  +  tan'  dx"  ... 

z  =  tan  6x ' — -r ,     ....     (184) 

or 


tan  e  = (185) 


The  point  (x,  z)  can  therefore  be  reached  by  one  or  by  two 
angles  of  projection,  according  as  the  quantity  under  the  radical 
sign  is  zero  or  positive,  and  cannot  be  reached  when  this  quan- 
tity is  negative. 

It  is  evident  that  if  the  parabola  whose  equation  is 

4>^'  -  4/^0  -  ^»  =  o (186) 

be  revolved  about  its  axis  2,  it  will  generate  a  surface  which  will 
be  the  locus  of  all  points  which  can  be  reached  by  but  a  single 
angle  of  projection,  and  beyond  which  the  projectile  cannot  be 
thrown  by  the  initial  velocity  due  to  h.  Any  point  within  this 
surface  can  be  reached  by  two  angles  of  projection.     This  limit- 


94  MECHANICS  OF  SOLIDS. 

ing  surface  is  a  paraboloid  of  revolution,  whose  vertex  is  on  the 
axis  of  z  at  a.  height  /i  from  the  origin,  and  the  radius  of  whose 
circular  section  in  the  plane  normal  to  the  axis  through  the 
origin  is  2^, 

94.  (4)  T/ie  Trajectory  in  Air. — The  air  resists  the  passage  of 
a  projectile  through  it,  and  thus  abstracts  a  part  of  its  kinetic 
energy.  The  resistance  of  the  air  is  chiefly  due  (i)  to  the  dis- 
placement of  the  air  particles  by  the  forward  motion  of  the  pro- 
jectile; (2)  to  the  excess  of  air  pressure  in  front;  (3)  to  the  cohe- 
sion of  the  air  and  its  friction  on  the  surface  of  the  projectile. 
The  resultant  of  these  forces,  called  the  resistance  of  the  air,  there- 
fore varies  with  the  velocity  and.  form  of  the  projectile,  the  nature 
of  its  motion,  and  the  condition  of  the  atmosphere.  It  is  a  force  of 
variable  intensity,  and  its  law  of  variation  is  not  accurately 
known;  hence  the  theorem  of  energy  Eq.  (126)  cannot  be  directly 
applied,  since  X,  V,  Z  are  unknown. 

When  the  velocity  of  any  particular  projectile  is  known  at 
certain  points  of  its  trajectory,  the  loss  of  energy  between  any 
two  of  these  points,  making  due  allowance  for  the  effect  of  known 
forces,  will  be  iM(  V^  —  V^),  which,  being  divided  by  the  dis- 
tance between  the  two  points,  will  give  the  mean  resistance.  An 
approximate  law  of  resistance  may,  be  obtained  by  taking  these 
points  sufficiently  near  together,  and  then  varying  the  initial 
velocity  so  as  to  include  all  service  velocities.  The  details  of 
this  method  being  given  in  the  course  of  Ordnance  and  Gunnery, 
only  a  brief  statement  of  the  mechanical  principles  is  here  given. 

Consider  the  projectile  to  have  motion  of  translation  only, 
and  the  acting  forces  to  be  the  weight  and  the  resistance  of  the 
air.  Let  W  be  the  weight  in  pounds,^  the  acceleration  due  to 
gravity,  r  the  acceleration  due  to  the  resistance  of  the  air,  Fthe 
velocity  of  the  centre  of  mass  at  any  point  of  the  trajectory,  and 
0  the  angle  which  the  trajectory  makes  with  the  axis  of  x.  Since 
the  resistance  of  the  air  acts  along  the  tangent,  the  trajectory 
will  lie  in  a  vertical  plane,  and  the  axes  of  co-ordinates  may 
therefore  be  assumed  as  in  Art.  93. 

The  total  acceleration  at  any  point  will  be  the  resultant  of 


GENERAL  THEOREM  OF  ENERGY.  95 

two  accelerations,  one  in  the  direction  of  the  tangent  to  the  tra- 
jectory and  the  other  in  the  direction  of  the  radius  of  curvature, 
Eq.  (8),  Art.  16.     These  are 

•^  =  -r-'gsin<p (187) 

and  ^ 

=— ^COS0,     .      .     ..      .      .      .      .       (188) 

H 

respectively. 

The  component  acceleration  in  the  direction  of  the  axis  of 
X  will  be  independent  of  gravity,  and  that  along  the  radius  of 
curvature  will  be  independent  of  the  resistance  of  the  air.  The 
component  velocities  along  x  and  z  are 

dx 

■^-  =  V cos  (f)  =  u; (189) 

dz 

-j-  =  F  sin  <p  =  u  tan  0 (^9°) 


The  acceleration  along  x  is  evidently 


j.  =  -rcos(/>, (191) 

whence  we  have 

du 

dt= -; (192) 

r  cos  <p  \  ^  / 

dx  = -; (193) 

r  cos  0  ^  ^"-^^ 

and  since 

dz  =  dx  tan  0, (194) 


96 


MECHANICS   OF   SOLIDS. 


we  have 


dz  = 


u  tan  (f)du 
r  cos  0 


(195) 


The  component  acceleration  along  the  radius  of  curvature, 

ds 
when  for  p  its  value  -r-:  is  substituted,  becomes 
dq> 


V^       ,,ds    I        ..ds  dcf)      ..d(t>  .  i     ^\ 


and  therefore 


^<f>=-^^f  =  ^. (.97) 


From  Eqs.  (192),  (193),  (195),  (197),  we  have 


0 


du 


cos  0' 

udu 


cos  0 

2^  tan  (})du 


r  cos 


ngdu_ 
J    Vr 


(198) 


These  integrations  depend  upon  the  variables  u^  0  and  r,  and 
as  there  is  no  known  relation  connecting  these  quantities,  the 
direct  solution,  which  requires  x,  z^  0and  /  to  be  known  through- 
out the  entire  trajectory,  is  impossible.  The  methods  of  ap- 
proximation used  in  the  solution  of  practical  problems  of  ballis- 
tics will  be  found  in  the  course  of  Ordnance  and  Gunnery. 


MOTION  OF  ROTATION,  97 


Motion  of  Rotation. 

95,  Moments  of  Inertia. — When  a  body  rotates  about  an  axis, 
the  velocity  of  any  molecule  is  (Art.  17) 

v—oor, (199) 

r  being  its  distance  from  the  axis.  Its  kinetic  energy  of  rotation 
is  therefore 

^mv^  =  \moo^r^^ (200) 

and  that  of  the  whole  body  is 

^'2mv^  =  ioo*2m.r*j (201) 

or  one  half  the  product  of  the  square  of  the  angular  velocity  by 
2mr*.  The  latter  is  called  the  moment  of  inertia  of  the  body  with 
respect  to  the  axis,  and  is  the  sum  of  the  products  obtained  by  multi- 
plying the  mass  of  each  molecule  by  the  square  of  its  distance  from  the 
axis. 

Since  for  a  given  angular  velocity  of  the  body  about  different 
axes  the  kinetic  energy  of  rotation  is  directly  proportional  to 
'2mr^^  the  moment  of  inertia  of  a  body  measures  the  capacity  of  the 
.body  to  store  up  kinetic  energy  during  a  motion  of  rotation  about  the 
axis  with  respect  to  which  the  moment  of  inertia  is  taken. 

The  angular  velocity  being  the  actual  velocity  at  a  unit's  dis- 
tance from  the  axis,  we  may  write 


or 


\M ,QD^  =^  \Q!)^'2mr\ (202) 

M^  =  J2/«r' (203) 


Hence,  ^wr'  measures  the  mass  which  would,  if  concentrated  at  a 
unifs  distance  from  the  axis,  have  the  same  ?fwment  of  inertia  as  the 
body  with  respect  to  that  axis. 

7 


98 


MECHANICS   OF  SOLIDS. 


96.  Radius  and  Centre  of  Gyration. — Let  M  be  the  mass  of  the 
body,  and  write 

^'  =  '2mr'^ (204) 


Solving  with  respect  to  ^,  we  have 
k 


=  V' 


^mr"^ 


M 


(205) 


The  distance  k  is  called  a  radius  of  gyration^  and  its  extremity 
not  on  the  axis,  a  centre  of  gyration.  When  the  axis  passes 
through  the  centre  of  mass  the  radius  and  centre  of  gyration  are 
coWtd  principal,  and  such  a  radius  is  generally  denoted  by  k^. 

From  Eq.  (204)  we  see  that  //  the  whole  mass  of  the  body  be  con- 
centrated at  the  centre  of  gyration  the  moment  of  inertia  of  the  body 
with  respect  to  the  axis  will  not  be  changed.  The  radius  of  gyration 
may  therefore  be  defined  to  be  the  distance  from  the  axis  at  which 
the  whole  mass  of  the  body  may  be  concentrated  without  changing  its 
moment  of  inertia. 

Since  the  kinetic  energy  of  rotation  of  a  body  depends  only 
on  the  angular  velocity  and  moment  of  inertia,  we  see  that  for  a 
given  value  of  qd  the  kinetic  energy  of  rotation  is  the  same  as  if 
a  mass  equal  to  2mr^  were  concentrated  at  a  unit's  distance  from 
the  axis,  or  the  whole  mass  of  the  body  at  the  distance  k  from 
the  axis. 

97.   The  Momental  Ellipsoid. — Let  it  be  required  to  find  the 
relations  existing  between  the  moments  of 
inertia  of  a  body  with   respect  to  all  right 
^  lines  passing  through  a  single  point.     Let 
the  assumed  point  be  taken  as  the  origin 
(Fig.  42),  and  let  a,  /3,  y  be  the  type-sym- 
3  bols  of  the  angles  made  by  the  right  lines 
-     with  the  co-ordinate  axes.     Let  m  be  the 
mass  of  one  of  the  molecules  of  the  body, 
and  we  have  for  its  moment  of  inertia,  with 
respect  to  OR, 


Fig.  42. 


mr"^  =  m[x^  -\- y^  -\-  z^  —  (x  cos  a  ^^^^  cos  /3  -\-  z  cos  yY],  (206) 


MOTION   OF  ROTATION. 


99 


Summing  the  moments  of  inertia  for  all  the  molecules,  we 
have,  for  the  moment  of  inertia  of  the  body  with  respect  to  OR^ 

'Smr^  —  ^m^x^  +  /  +  z^  —  (^^  cos  or  +  /  cos  /3  +  2  cos  yY\ 

=  2m{{x'^-\-y'^-\-  z'^){cos^a  +  cos^  /?-j-  cos^  r)—{x  cos  a  -\-y  cos  /?+2  cos  yf^ 

-  2m{f  +  z'')  cos'  a  +  2m{x''  +  z^)  cos^  /3  +  2m{x''  +/)  cos'  r 

—  2^myz  cos  ^  cos  y  —  22mxz  cos  «  cos  >^  —  22mxy  cos  a  cos  /?.  .     (207) 

But  2m{y'  +  2'),  :S;//(^'  +  z')  and  ^/^^(jt:'  +/)  are  the  mo- 
ments of  inertia  of  the  body  with  respect  to  the  axes  x,  y,  Zy 
respectively.  Representing  these  moments  of  inertia  by  A,  B,  C, 
we  have 

2mr^  =  A  cos'  a  -{•  B  cos''  /3  -\-  C  cos'  y  —  22myz  cos  /3  cos  y 

—  22mxz  cos  a  cos  ;^  —  22mxy  cos  of  cos  jS.  (208) 


Lay  off  on  OR  a  distance  from  6>  equal  to 


f^ 


:,  and  let 


mr 


x',  y,  0'  be  the  co-ordinates  of  the  point  thus  determined.  Then 
we  have 


or 


*'  = 

cos  a 

V2mr'' 

/= 

cosyS 

z'  = 

cos  y 

cos  a  = 

x'  V2mr' 

cos/?  = 

y  V:Emr^ 

(209) 


cos  y  =^  z^  4/^ 


(210) 


Substituting  these  values  of  the  cosines  in  Eq.  (208),  we  have 

.     (211) 


I  =  Ax''  +  By''  +  a"  -  2{^2myz)yz'  -  2{2mxz)x'z' 

—  2(2mxy)x'y' 


c 


ICXD  MECHANICS  OF  SOLIDS. 

which  is  the  equation  of  the  locus  of  all  points  that  are  at  a  dis- 
tance from  the  origin  equal  to  the  reciprocal  of  the  square  root 
of  the  moment  of  inertia  of  the  body  with  respect  to  the  line 
upon  which  this  distance  is  laid  off.  We  see  from  the  equation 
that  this  locus  is  a  surface  of  the  second  order;  and,  since  the 
radius-vector  is  always  finite,  it  is  an  ellipsoid. 

This  ellipsoid  is  called  the  momental  ellipsoid  of  inertia^  for  the 
reason  that  the  square  of  the  reciprocal  of  any  one  of  its  semi- 
diameters  is  the  moment  of  inertia  of  the  body  with  respect  to 
the  coincident  right  line.  It  presents  a  geometrical  image  of 
the  values  of  the  moments  of  inertia  of  the  body  with  respect  to 
all  lines  radiating  from  the  assumed  point. 

The  greatest  moment  of  inertia  is  that  with  respect  to  the 
shortest  diameter,  and  the  least  is  that  with  respect  to  the  great- 
est diameter.  As  all  semi-diameters  of  the  cyclic  sections  are 
equal  to  the  mean  semi-axis  of  the  ellipsoid,  the  moments  of 
inertia  with  respect  to  these  lines  are  equal  to  each  other.  The 
origin  O  having  been  assumed  at  pleasure,  it  is  evident  that 
there  is  a  momental  ellipsoid  of  the  body  for  each  point  in  space. 

98.  Principal  Axes. — Since  the  equation  of  an  ellipsoid  when 
referred  to  its  centre  and  axes  takes  the  form  of 

Ax*""  +  Bf^  4-  a'"  =   I, (2I2> 

we  see  that  for  at  least  one  set  of  rectangular  co-ordinate  axes 
through  any  point  in  space  we  must  have  the  conditions 


E^l 


2mxy  =  o; 

2mxz  =  o;  J- (213) 

2myz 


The  axes  of  figure  of  the  momental  ellipsoid  are  called  prin- 
cipal axes  at  the  point  considered;  and  since  for  such  axes  we 
have  the  conditions  expressed  by  Eqs.  (213),  the  latter  are  called 
l/ie  conditions  for  principal  axes. 


\i/K^\  vv* 


Qi  i    v^Y 


MOTION-  OF  ROTATION.  lOI 

The  quantities  2mxy,  2mxZy  2myz  reduce  to  zero  for  princi- 
pal axes  because  the  sum  of  the  positive  and  negative  products 
arising  from  the  signs  of  the  co-ordinates  x,  y,  z  are  numerically 
equal  to  each  other.  The  moments  of  inertia  for  such  axes  are 
called  principal  moments  of  inertia;  they  evidently  include  the 
greatest  and  least  moments  of  inertia  at  the  point. 

The  value  of  JS'/z/r^  for  any  axis,  Eq.  (211),  in  terms  of  the 
principal  moments  of  inertia  at  the  point  considered,  becomes 

^wr'  =  A  cos'  a-\-  B  cos'  ^ -\- C  cos'  y\  .     .     (214) 

that  is,  the  moment  of  inertia  of  any  body  with  respect  to  any  line  what- 
€7!er  is  equal  to  the  sum  of  the  products  obtained  by  multiplying  the 
principal  moments  of  inertia  at  any  point  of  the  line^  respectively^  by  the 
squares  of  the  cosines  of  the  angles  which  the  line  makes  with  the  prin- 
cipal axes  at  the  point. 

99.  It  is  readily  seen,  Eq.  (214),  that  v^rhen  the  principal  mo- 
ments of  inertia  of  a  body  are  known  at  any  point,  all  its  other 
moments  of  inertia  with  respect  to  that  point  may  be  deter- 
mined; and  it  will  now  be  shown  that  if  the  principal  moments 
of  inertia  be  known  at  the  centre  of  mass,  the  moments  of  inertia 
with  respect  to  all  lines  whatever  can  be  readily  computed. 
Let  any  right  line  be  taken  as  the  axis  of  z\  then  the  moment  of 
inertia  with  respect  to  this  line  is 

2/«r'  =  ^/«(^'+/) (215) 

Let  ^o>^o  ^^  the  co-ordinates  of  the  centre  of  mass  referred 
to  the  assumed  axes,  and  x\y*  the  co-ordinates  of  the  molecules 
of  the  body  referred  to  the  centre  of  mass;  then  we  have 

x=zx^-\-x*\   y=y,+y'; 

which,  substituted  in  Eq.  (215),  give 

Smr'  =  S>n{x'  +/)  =  2m[{x,  +  x')'  +  {y,  +/)'] 

=  2m{x,'+y;)  +  2m{x"+/')+2x,2mx'+2y,2m/.  (216) 


I02  MECHANICS  OF  SOLIDS. 

Placing  x^  -\-  y^  =  ^',  and  remembering  that  by  the  princi- 
ple of  the  centre  of  mass 

^mx'  =  ^my'  =  o, 
we  have 

2mr'=:Md'^:2m{x''' -{■/') (217) 

Therefore,  the  moment  of  inertia  of  the  body  with  respect  to  any 
line  in  space  is  equal  to  its  moment  of  inertia  with  respect  to  a  parallel 
line  through  the  centre  of  mass,  increased  by  the  product  of  the  mass  of 
the  body  by  the  square  of  the  perpendicular  distance  from  the  centre  of 
mass  to  the  given  line.  The  least  principal  moment  of  inertia  at 
the  centre  of  mass  is  therefore  the  least  of  all  the  moments  of  in- 
ertia of  the  body. 

100.  Discussion  of  the  Momental  Ellipsoids  of  a  Body. — Let  Ay 
£,  C  be  the  principal  moments  of  inertia  at  the  centre  of  mass. 

(i)  Suppose  A  =^  B  ^^  C.  Then  the  central  ellipsoid  is  a 
sphere,  and  therefore  all  moments  of  inertia  at  the  centre  of 
mass  are  equal,  and  all  axes  through  it  are  principal.  For  every 
other  point  in  space  the  ellipsoid  is  a  prolate  spheroid  whose 
axis  passes  through  the  centre  of  mass;  for,  the  moment  of  in- 
ertia with  respect  to  this  line  is  the  same  as  the  central  mo- 
ments of  inertia,  while  those  with  respect  to  all  lines  perpendicu- 
lar to  this  are  greater  than  the  central  moments  of  inertia  and 
equal  to  each  other;  and  these  lines  are  principal  axes  since  the 
moment  of  inertia  with  respect  to  the  line  through  the  centre 
of  mass  is  the  least  of  all  the  moments  of  inertia  at  the  point  in 
question. 

(2)  Suppose  A  y  B  and  B  =.  C.  The  central  ellipsoid  is  an 
oblate  spheroid  whose  axis  is  that  of  the  greatest  moment  of 
inertia.  There  are  two  points  on  the  axis  of  the  spheroid  at 
which  the  ellipsoid  is  a  sphere,  and  they  are  found  thus:  At 
these  points  all  moments  of  inertia  must  be  equal  to  A.  Then, 
denoting  by  x  the  distance  of  these  points  from  the  centre,  we 
have 


MOTION  OF  ROTATION.  IO3 

A=iB-^Mx^^C^Mx^',      ....     (218) 
whence 


lA-B 


(219) 


It  is  evident  that  the  ellipsoid  can  be  a  sphere  at  no  other 
point. 

(3)  Suppose  A  =  B  and  B  >  C.  The  central  ellipsoid  is  a  'Tlt~" 
prolate  spheroid   whose  axis  is  that  of  C.     There  is  no  point  at 

which  the  ellipsoid  can  be  a  sphere. 

(4)  When  A  >  B  >  Cy  the  central  ellipsoid  is  one  of  three  ^  /^  ^  ^ 
unequal  axes  at  the  centre  of  mass,  and  cannot  be  a  sphere  at  ^.^ 
any  point  in  space.                                                                                           )^'\^. 

10 1.  Determination  of  the  Moment  of  Inertia. — The  moment  of      ^^  1,^^ 
inertia  may  sometimes  be  found  by  the  summation  of  the  sepa-         ' 
rate  values  of  wr*. 

Whenever  the  body  is  one  whose  density  and  boundary  vary 
by  some  law  of  continuity,  we  may  write 

m-=  dAf  =■  6dV (220) 

and 

2mr^=rr^dM=rr''6dV', (221) 

from  which  the  moment  of  inertia  can  be  found  whenever  the 
expression  can  be  integrated  between  the  limits  that  determine 
its  volume. 

Having  found  the  moment  of  inertia  of  a  body  with  respect 
to  any  line,  that  with  respect  to  a  parallel  line  may  be  found  by 
Eq.  (217);  and  having  found  the  principal  moments  of  inertia  at 
any  point,  that  with  respect  to  any  other  line  through  the  point 
can  be  found  from  Eq.  (214). 

In  the  following  examples  let  S  represent  the  density,  go  the 
area  of  cross-section  of  a  material  line,  and  /  the  thickness  of  a 
material  surface. 


104  MECHANICS  OF  SOLIDS. 

Ex.  I.  A  Uniform  Straight  Rod. — Let  the  axis  be  perpen- 
dicular to  the  rod  at  its  middle  point,  and  represent  the  length 
of  the  rod  by  2a\  then  we  have 

x^dx  =  2a6oo— . 
3 

Whence,  since  M  =  2adGo,  we  have 


Mk;  =  M-     and     k/ =  - (222) 

3  '3  ^ 


The  centre  of  gyration  is  therefore  at  a  distance  from  the 

centre  of  mass  equal  to =  .577^:,  nearly. 

1.732  -I-        *^"   ' 

For  an  axis  perpendicular  to  the  rod  at  any  distance  d  from 

the  centre  we  have 


)mr 


=  Ml+'^')= ("3) 


which  becomes,  for  the  perpendicular  axis  at  either  extremity. 


M±a\ 


Ex.  2.  A  Circular  Arc^  subtending  an  angle  2^  at  the  centre 
and  whose  radius  is  a, 

I.  Axis  perpendicular  to  the  plane  of  the  arc  through  its 
centre. 

■:2mr''  =  doo  f^a'dO  =  26000,' 6  =  Ma*) 


MOTION  OF  ROTATION.  IO5 

and,  for  the  whole  circumference,  ' 

^wr'  =  2716000"  =  Ma* (224) 

2.  Axis  in  the  plane  of  the  arc  through  its  centre  and  middle 
point. 

Smr"  =  doo  C^  a'  sin'  Odd 

=  dooa\d  —  sin  6  cos  0)\ 

and,  for  the  whole  circumference, 

Smr"  =  TtSaoa*  =  J/ -. (225) 


From  Eqs.  (224)  and  (225)  2mr*  can  be  found  for  any  right 
line  passing  through  the  centre  by  the  application  of  Eq.  (214). 

.Ex.  3.  A  Rectangular  Plate  whose  sides  are  2a  and  2b.  Take 
the  centre  of  the  plate  as  the  origin,  and  the  axes  parallel  to 
its  sides;  then  for  the  axis  x^  perpendicular  to  2^,  we  have 
dM  =  2adtdy^  and  therefore 

2mrJ  =  2a6t  f^ y'dy  =  '^6tab'^M^^\    .     .     (226) 
J-b  Z  3 

and  similarly,  for  the  axis  y^ 

2mry  =  M" (227) 


For  the  axis  z  we  have 

2m(x*  +/)  =  2m(y'  +  z")  +  2m{a^  +  2"), 


io6 


MECHANICS   OF  SOLIDS. 


since  z  =  o;  hence 


2Mr/=.M^^±^ (228) 

3  ^       ' 


If  the  plate  be  square^  a  =  b,  and  we  have 


'2mrJ  =  SmrJ'  =  M 


:^mr^  =  M 


y 
2a' 


(229) 


Ex.  4.  A  Triangular  Area  about  an  Axis  through  a  Vertex. — 

V     ^ B  Let  AjBC,  Fig.  43,  be  the  triangle,  and  take 

the  axes  x  and  y  through  the  vertex  in  its 
plane;  let  /3  and  yS'  be  the  distances  of  the 
vertex  B  from  y  and  x  respectively,  and  y 
and  ;/'  those  of  the  vertex  C  from  the  same 
axes,  and  let  AZ>  =  /;  then  we  have  for  the 
triangle  ABD 


Fig.  43. 


dM  =  dtPQdx  -  dtl^—^dx. 


Similarly  we  have,  for  the  triangle  ABC, 

:Emry'  =  dtl  r(i  -  -\x''dx  =  (J//^-.  .      .     (231) 


But  ^mry  of  ABC  is  equal  to  the  difference  betw-en  that  of 
ABB  and  ACD\  therefore  we  have,  for  ABC, 


MOTION  OF  ROTATION.  lO/ 

:2Mr;  =  Stl^^ (,3,) 


The  mass  of  ABC  is  evidently  dtl— ^;  hence 

2 

:2mr;  =  ~l,P' ^  §Y  +  Y') (^33) 

which  is  a  general  formula  for  the  moment  of  inertia  of  any  tri- 
angle with  reference  to  an  axis  in  its  plane,  passing  through  its 
vertex  and  being  wholly  without  the  triangle.    Similarly  we  have 

:S«r,'  =  ^(/J" +  /?>'  +  /'),.     .     .     .     (234) 

and,  for  the  axis  z  at  A, 

2mr.'  =  ^(/J'  +  fiy  +  y'  +  P"  +  /?>'  +  y").  .    (23s) 


Ex.  5.  A  Triangular  Area  about  any  Axis  whatever. — At  the 
middle  point  of  each  side  let  one  third  the  mass  of  the  triangle 
be  concentrated;  then  the  centre  of  gravity  of  the  three  material 
points  coincides  with  that  of  the  triangle. 

The  moment  of  inertia  of  the  material  points  with  reference 
to  any  line  Ay  drawn  through  A  is 

and  with  reference  to  a  line  through  A    perpendicular  to  the 
plane  of  the  triangle  is 


108  MECHANICS  OF  SOLIDS. 


M 
3 


Uf+f)+(?+?)+('^+'^^)( 


=  -^W  +  Py  +  f  +  r  +  P'y'  +  r");  (^'s?) 


and  these  moments  of  inertia  are  the  same  as  those  of  the  tri- 
angle with  respect  to  the  same  lines. 

The  moments  of  inertia  expressed  by  Eqs.  (235)  and  (237) 
are  evidently  the  greatest  moments  of  inertia  at  the  point,  and 
they  are  therefore  principal  moments  of  inertia;  hence  two  of 
the  principal  axes  lie  in  the  plane  of  the  triangle.  But,  Eqs. 
(233)  and  (236),  the  sections  by  this  plane  of  the  ellipsoids  for 
the  point  A  of  the  triangle  and  system  of  points  coincide 
throughout  at  least  half  their  length;  therefore  the  principal 
axes  in  this  plane  are  also  equal  and  coincident,  and  the  ellip- 
soids coincide  throughout.  Hence  the  ellipsoids  at  the  common 
centre  of  mass  are  one  and  the  same  ellipsoid.  The  triangle  and 
system  of  material  points,  having  then  the  same  central  ellipsoid, 
have  equal  moments  of  inertia  with  respect  to  all  lines  in  space. 
Therefore,  to  determine  the  moment  of  inertia  of  a  triangular 

M 
area,  find  that  of  three  masses  each  equal  to  —  at  the  middle  of 

its  sides  with  respect  to  the  given  line,  and  it  will  be  the  required 
moment  of  inertia  of  the  triangle. 

Ex.  6.  An  Elliptical  Area. — Let  the  equation  of  the  ellipse  be 

ay  +  ^V  =  a^V"', 

then,  for  a  line  through  its  centre  coincident  with  its  major  axis, 
we  have 


]mrx  ^  =■  461  j     j    y^dxdy 

t/o     t/o 

=  7tabdt-=M-- 
4  4 


'.     .     (238) 


'^^tfCiT'-^;: 


MOTION  OF  ROTATION.  IO9 

after  substituting -(«'  —  x"^  iox- y  after  the  first  integration. 

Oi 

Similarly,  for  the  minor  axis,  we  have 

2mry  =  ^6t  I     j    x^dydx 

t/o     t/o  I  .  . 

«»   r  •  •  •  ^'^9) 

4  4        J 

and  by  combination  we  have 

2».r/  =  ^^^l±^) (240> 

For  an  axis  in  the  plane  of  the  ellipse  coincident  with  any  ra- 
dius vector  r  we  have,  Eq.  (214),  since  y  =  r'  sin'  a  and  x^  =  r* 
cos'  a, 

2mrr^  =  M  ("'  ^^"'  ^  +  ^'  '""^^  =  M^.    .     (244 
4  4r'  ^  ^  ' 

For  a  circular  area  we  have 


4  2 

Ex.  7.  ^«  Ellipsoid. — Let  the  equation  of  the  ellipsoid  be 

then  the  area  of  any  section  perpendicular  to  the  axis  of  x  is 

a  a  a   ^  ' 

and  therefore 


no 


MECHANICS  OF  SOLIDS. 


dM  =  d^{a'  -  x^)dx. 


From  Eq.  (240)  we  see  that  the  square  of  the  radius  of  gyra- 
tion for  an  ellipse  with  respect  to  the  normal  through  its  centre 
is  equal  to  the  sum  of  the  squares  of  its  semi-axes  divided  by  4; 
therefore  we  have 


•'^  nbc 


./'  +  ^%.. 


:2mr,'  =  S  r   "^{a^  -  xy-—^-(a'  -  x*)dx 


^^jc(^-^n  r,^. 


Sn^^^     \'l  r  (a'  ~  xy. 

-u  Tiauc 
3 


dx 


—dnabc 


5 


=  M 


b'  +  c' 


In  the  same  way  we  readily  obtain 

•"                  5      ' 
^mr^   =  M ' — . 


(242) 


(243) 


For  a  spheroid  whose  axis  is  a,  and  b  =■  c,  vjq  have 


^pirj"  =  M 


2a' 


'!2mry   =  "^mrz  =  M 


a^-\-b'' 


(244) 
(245) 


For  a  sphere^ 


2ar 


:SmrJ  =  2mry^  =  2mr/  =M — . 

0 


(246) 


MOTION  OF  ROTATION. 


Ill 


Ex.  8.  A  Rectangular  Par allelopipedon, — Assume  the  origin  at 
one  of  the  angles  of  the  solid;  let  the  co-ordinate  axes  coincide 
with  its  edges,  and  let  a,  b,  c  be  their  lengths  along  the  axes  x^ 
y,  Zf  respectively;  then 


A=  r  f^  r p{f  +  z^)dxdydz 

«/o     t/o    t/o 

__  pabcip'  -f  c*)  __  ^^^'  +  c\ 
3  3 

^  ^  pabc{a' -\- c')  ^  ^^l+f\ 
3  3      ' 

^  ^  pabcia^-^b-^)  ^  j^a^jV_b\, 
3  3      ' 


and  for  principal  axes  at  the  centre  of  mass,  (Art.  99), 


A^m'L±S\     B=.M^l±fl     C=M^1±^:, 


(247) 


12 


12 


12 


(248) 


For  the  cube^  since  a=z  b  =  c.vjq  have  for  the  edges 


2«' 


A=B=C= M^ 


and  for  principal  axes  at  the  centre  of  mass 

\   -i  2   /  6 


(249) 


Hence  the  momental  ellipsoid  at  the  centre  of  the  cube  is 
a  sphere,  and  all  moments  of  inertia  are  equal. 

It  might  appear  that  the  edges  are  principal  axes  at  the  angle 
of  the  cube  from  the  equality  of  their  moments  of  inertia.  But 
the  line  joining  the  centre  and  angle  is  a  principal  axis  at  the 
angle,  since  the  moment  of  inertia  with  respect  to  it  is  less  than 
that  with  respect  to  any  other  line;  hence  the  principal  axes  at 


112 


MECHANICS  OF  SOLIDS. 


the  angle  are  the  diagonal  of  the  cube  and  any  pair  of  perpen- 
dicular right  lines  in  the  normal  plane  to  the  diagonal  at  the 
angle.  The  moment  of  inertia  with  respect  to  every  right  line 
in  this  plane  passing  through  the  angle  of  the  cube  is  readily 
seen  to  be 


M 


(6+t)  =  ^— (^5°) 


102.  The  moments  of  inertia  with  respect  to  central  principal 
axes  are  here  tabulated  for  convenient  reference. 


Mass. 


Dimensions. 


X. 


Y. 


1.  Rod  or  Cylinder 

2.  Circular  Rim 

3.  Rectangular  Plate 

4.  Elliptical  Area 

5.  Circular  Area 

6.  Ellipsoid 

7.  Spheroid 

8.  Sphere 

9.  Rectangular  Paral 

lelopipedon 

10.  Cube 


length  =  2a, 
radius  =  r 

radius  =  a 
sides=aand^ 
axes=«and3 
radius  =  a 
axes  =  a,  d,  c 
d  =  c 

radius  =  a 
edges  =  a,d,c 
edge  =  a 


M  - 
3 

Ma" 


4      3 


AI 


b'' 


M- 


M 


M 


M' 


4 

5 
5 

2^2 


h'^ 


■M 
4  3 

-M  — 
2      3 


M  — 
3 

Md" 


M- 
4      3 


.a'' 


2 
M- 
4 


a'- 
4 

5 

5 

M  — 
5 

-M ■ 

4  3 


-M- 
2       3 


M- 
2 

M- 
2 


2 


5 


4  3 

-M  — 
2       3 


From  the  moments  of  inertia  above  we  can  readily  derive  the 
corresponding  radii  of  gyration  by  dividing  by  the  mass  of  the 
body  and  taking  the  square  root  of  the  quotient. 


INSTANTANEOUS  AXIS.  II3 

If  the  body  be  irregular  in  form  and  be  not  homogeneous, 
the  principles  of  the  calculus  cannot  be  applied  to  find  its  mo- 
ment of  inertia.  In  such  cases  the  moment  of  inertia  can  be 
experimentally  found  by  means  of  the  principles  of  the  com- 
pound pendulum,  a  method  which  will  be  explained  subse- 
quently. 

Instantaneous  Axis. 

1O3.  Whatever  may  be  the  component  angular  velocities  of 
a  body  when  rotating  about  a  centre,  the  resultant  angular  ve- 
locity and  the  axis  of  rotation  may  be  found  by  the  application 
of  the  principle  of  the  parallelopipedon  of  angular  velocities.  4? 
When  the  centre  of  mass  is  taken  as  the  centre  of  rotation,  these 
are  called  instantaneous  angular  velocity  and  instantaneous  axis.  At 
any  instant  the  path  of  each  molecule  of  the  body  is  in  a  plane 
perpendicular  to  the  instantaneous  axis,  and  all  points  on  this 
axis  have  no  motion  with  respect  to  the  centre  of  mass. 

The  component  velocities  of  any  molecule  with  respect  to 
the  centre  of  mass  are  obtained  by  dividing  Eqs.  (116)  by  d?"/; 
thus  we  have  -    ^ 


dx'        ,dtb         ,  d(b        ,  , 


dt  ~~  dt      -"    dt  "^      -^     '' 

d/         /0        ,dd         , 
■^="-^-^^="^^-^^- 
dz'        ,dd  Jtb 

dt         ^     dt  dt         ^      ""  •^' 


(251) 


when  we  substitute  the  symbols  oox^  ooy^  oo^  for  the  component 

,     .  .      dd    dtp    d(l)     ^ 
angular  velocities  — ,  -^,  -j-   about  the  co-ordinate  axes  x\yy 

z',  respectively.     If  in  these  equations  we  make 

dx^__^  _  dz^_ 
dt  ^  dt  ~  dt  "^  °' 


114  MECHANICS   OF  SOLIDS. 

we  shall  obtain  the  equations  of  the  locus  of  all  those  points 
which  are  at  rest  with  respect  to  the  centre  of  mass;  and,  since 
these  points  lie  on  the  instantaneous  axis,  we  have 

z'ooy  —  y'coz  =  o,  ) 

X^GOz  —  Z^GDje=  0,y (252) 

y'GJj;  —  x'ooy  —  o,) 

for  the  equations  of  the  instantaneous  axis.  All  molecules  of 
the  body  not  on  this  right  line  will  have,  at  the  assumed  instant, 
a  motion  with  respect  to  the  centre  of  mass,  and  will  therefore 
rotate  about  it.  Since  this  axis  passes  through  the  centre  of 
mass,  the  position  of  the  axis  in  space  depends  only  on  the  val- 
ues of  the  angular  velocities  gDjc,  cOy,  gOz;  and  as  these  values 
generally  remain  constant  only  for  the  instant  df,  the  instanta- 
neous axis  describes  the  surface  of  a  cone  whose  vertex  is  the 
centre  of  mass. 

104.  Let  a,  /3f  y  he  the  angles  which  the  instantaneous  axis 
makes  with  the  co-ordinate  axes,  and  co  the  instantaneous  angu- 
lar velocity;  then,  by  the  principle  of  the  parallelopipedon  of  an- 
gular velocities,  we  have 

COS  a  =  — ,     cosp  =  -^,    COS  V  =  — ,      .     (253) 

00  '  00  '  00 

and 

co'  =  G?^'  -f  (W^  '  4-  ffi>^  » (254) 

Hence  it  is  necessary  to  find  values  for  the  component  angu- 
lar velocities  before  the  position  of  the  axis  and  the  resultant 
angular  velocity  of  the  body  can  be  determined.  The  two  cases 
to  consider  are  (i)  rotation  due  to  the  action  of  incessant  forces ^  and 
(2)  rotation  due  to  impulsions.  The  latter,  being  the  simpler  case, 
will  be  discussed  first. 


ROTATION  DUE    TO  IMPULSIVE  FORCES.  II5 


Rotation  of  a  Rigid  Solid  due  to  Impulsive  Forces. 

105.  When  the  centre  of  mass  is  the  movable  origin,  the  mo- 
tion of  translation  of  this  point  and  the  motion  of  rotation  of  the 
body  about  it  have  been  shown  to  be  wholly  independent  of 
each  other.  Hence  we  may  regard  the  centre  of  mass  as  a  fixed 
point  in  space,  and  consider  only  the  rotation  of  the  body  about 
it.  The  body  is  supposed  to  have  been  subjected  to  a  system 
of  impulsions  whose  effect  is  completed  in  a  very  short  time,  and 
the  body  is  then  abandoned  to  itself,  free  from  the  action  of  any 
extraneous  force  whatever.  It  is  required  to  find  its  subsequent 
motion  of  rotation  in  all  of  its  particulars. 

When  the  moments  of  the  several  impulsions  may  be  com- 
pounded into  a  single  resultant  moment,  Rk^  having  the  centre 
of  mass  as  the  centre  of  moments,  the  moment  axis  of  R  is  call- 
ed the  resultant  or  invariable  axis,  and  the  plane  of  R  and  k  is 
called  the  resultant  or  invariable  plane.  They  cannot  change  their 
direction  in  space  unless  other  forces  be  introduced,  which  is 
not  supposed. 

106.  Assuming  Eqs.  Rm',  which  are  here  applicable,  /  v  /  /  i> 

21  Xx'  cos  fi^y  COS  a)  =  :2v{^-^^^^^-^  =  Z„ 

2/,(z'  COS  a^x'  cos  y)  =  2m?^^^^^-^^f^  =  M,,    )-  {RJ) 

2lXy  cos  y^z'  cos  /?)  =  :2m^l^^-^I-^^  =  N,, 

and  substituting  the  values  of  dx\  dy\  dz'  given  in  Eqs.  (251), 
we  have 

:2m''''^^'~j'^'''  =^:2m(x''^y')cso,---:2mx'z'oD^--:2myz'GOy', 

z'dx* x'dz' 

'2m — =  2m(x'*-\-  z'^)G0y'-'2my^ z' GDs—'2mx'y' GDjc'y  K^SS) 

2m- — —  =  2m{y^-\-  2')<»^  —2mx'y'G0y—2mx'z'G0g, 


ii6 


MECHANICS   OF  SOLIDS. 


If  the  axes  be  principal,  these  equations  reduce  to 


/v^'        ^'^«' 


2V;c' 


^/ 


=  ^;??(y'  +  z'^)G9y   =  ^GJy  =  M,\ 


2/w^i^^^^'  =  :^/«(/»  +  2'»)a?,  =  ^(i?^  =  iV^^. 


(256) 


Hence 


Z,  i^,  ^.  ,      , 


That  is,  //z^  angular  velocity  due  to  an  impulsion  about  a  principal  axis 
is  equal  to  the  component  moment  of  the  impulsion  divided  by  the  moment 
of  inertia  of  the  body^  both  taken  with  respect  to  that  axis. 

This  principle  is  true  also  for  the  instantaneous  axis;  for  if  z 
be  the  instantaneous  axis,  we  have 

Cb?^  z=  Ce?^  =  o,     and     00^  =  Ce?. 

And  substituting  these  values  in  the  first  of  Eqs.  (255),  we  have 

:2m'^^-^—  =  :^m(x''  +/')o^.  =  Cg?  =  Z,.  .     (258) 


Since  Eqs.  (Rm')  apply  to  rotation  about  a  fixed  point  or  a 
fixed  axis,  this  principle  is  likewise  applicable  to  both  of  these 
cases. 

107.  Let  "^mr"^  be  the  moment  of  inertia  with  respect  to  the 
instantaneous  axis,  and  0  the  angle  between  this  axis  and  the 
invariable  axis.     Then  we  have 


00  = 


Rk  cos  0 


-v-- 


(259) 


ROTATION  DUE    TO  IMPULSIVE  FORCES.  11/ 

hence 

,     .         '2mv'^  =  G0^'2mr^  =  —  =  Ekoo  cos  0,  .     .     .     (260) 

d  being  that  semi-diameter  of  the  central  ellipsoid  which  coin- 
cides with  the  instantaneous  axis. 

Squaring  Eqs.  (256)  and  adding,  we  have 

A^QD:c'  +  B'ooy'  +  c'ci?^'  =  l;  -f  m;  +  n;  =  R^k*-,  (261) 

-ff-^  being  the  resultant  moment  of  the  system  of  impulsions  with 
respect  to  the  centre  of  mass. 

From  Eq.  (260)  we  conclude,  since  2mv^  is  constant, 

(i)  The  instantaneous  angular  velocity  varies  directly  with 
the  length  of  the  semi-diameter  of  the  ellipsoid  which  coincides 
with  the  instantaneous  axis,  or  inversely  as  the  square  root  of 
the  moment  of  inertia  with  respect  to  this  axis. 

(2)  The  angular  velocity  about  the  invariable  axis,  an  cos  0, 
is  constant;  therefore,  as  0  increases  or  cos  0  diminishes,  go  in- 
creases; that  is,  as  the  instantaneous  axis  increases  its  inclina- 
tion to  the  invariable  axis,  the  instantaneous  angular  velocity 
increases. 

Eqs.  (260)  and  (261),  together  with  that  of  the  central  ellip- 
soid, give  the  circumstances  of  the  rotary  motion  of  a  free  rigid 
solid  under  the  action  of  impulsive  forces  whenever  we  can  find 
the  value  of  Rk. 

108.  The  equation  of  the  invariable  plane  is 


Call  the  point  in  which  the  instantaneous  axis  pierces  the 
central  ellipsoid  the  instantaneous  pole,,  and  let  x\y',  z*  be  its  co- 
ordinates.    Then  the  equation  of  the  tangent  plane  to  the  ellip- 


Il8  MECHANICS  OF  SOLIDS. 

sold  at  the  instantaneous  pole  is 

Axx'  +  Byy*  +  Czz'  =  i.     .     .    .    .    .     (263) 

From  Eqs.  (252)  we  liave 

QD:c  OOy  CDz  GO  /    ,    v 

^  =  y=ir=d=^'    •  •  •  •  (^^4) 

which  reduces  Eq.  (263)  to 

AoOxX  +  Booyy  +  CoOzZ  =  €,  .     .    .     .     (265) 
and  this,  by  Eqs.  (256),  becomes 

N,x  +  M,y  +  Z,z=€ (266) 

Dividing  both  members  by  i?/^,  we  have 

The  sum  of  the  squares  of  the  coefficients  of  the  variables  be- 
ing unity,  these  coefficients  are  the  cosines  of  the  angles  which 
the  normal  to  the  plane  makes  with   the  co-ordinate  axes,  and 

—  is  the  perpendicular  distance  from  the  centre  of  mass  to  the 

plane.  We  therefore  see  that  the  tangent  plane  at  the  instanta- 
neous pole  is  parallel  to  the  invariable  plane,  and  that  these  two 
planes  are  separated  by  the  constant  perpendicular  distance, /. 

Hence  the  ellipsoid  ro//s,  without  sliding^  on  a  tangent  plane 
parallel  to  the  plane  of  resultant  rotation  of  the  system,  and  at  a 
fixed  distance/  from  the  centre  of  the  ellipsoid.  As  different 
points  of  the  ellipsoid  come  successively  into  the  tangent  plane> 


ROTATION  DUE    TO  IMPULSIVE  FORCES.  IIQ 

the  semi-diameters  which  join  them  with  the  centre  become  in 
turn  the  instantaneous  axis.  The  locus  of  the  tangent  points  on  the 
ellipsoid  is  called  the  Polhode^  and  in  the  general  case  is  a  curve 
of  double  curvature.  The  locus  of  the  points  of  contact  on  the 
tangent  plane  is  necessarily  a  plane  curve,  and  is  called  the  Her- 
polhode.  If  we  imagine  all  points  of  the  polhode  joined  with  the 
centre  of  the  ellipsoid  by  the  various  semi-diameters,  and  all 
points  of  the  herpolhode  with  the  same  point  by  the  various  in- 
stantaneous axes,  we  will  have  two  cones;  the  former  called  the 
rolling  cone,  described  about  a  principal  axis  of  the  ellipsoid,  and 
the  latter  called  the  directing  cone,  about  the  invariable  axis;  at 
any  instant  they  are  tangent  to  each  other  along  the  instanta- 
neous axis. 

109.   The  Rolling  Cone. — Dividing  both  members  of  Eq.  (261) 
by  e',  we  have 


^•^-  +  ^'^  +  C'^:-=j-...    .         .     (^68) 


But,  Eqs.  {264), 


GoJ         „       GoJ         ,,       00  * 


€' 


^=/»;      ^=Z'\     .      .      .      (269) 


and  Eq.  (268)  becomes 

^V"  +  ^y  +  C  V»  =  i,      ....     (270) 


which  is  an  equation  of  condition  for  points  of  the  polhode. 

Since  the  instantaneous  pole  is  on  the  ellipsoid,  we  have  also 
the  condition,  Eq.  (212), 

Ax''  +  B/'  +  a"  =  I, (27i> 


I20  MECHANICS   OF  SOLIDS, 

and  dividing  by/^  we  get 
> 

-Ti-+-^  +  -ir-=T, (272) 

/  /  /  / 

Subtracting  Eq.  (270)  from  Eq.  (272)  and  omitting  accents, 
we  have 

which  expresses  the  relations  existing  between  the  co-ordinates 
of  any  point  of  the  polhode.  But  we  see  that  this  equation  is 
satisfied  by  the  co-ordinates  of  the  origin,  and  also  by  any  set  of 
values  which  bear  a  constant  ratio  to  the  corresponding  co- 
ordinates of  any  point  of  the  polhode;  that  is,  the  equation  Is 
satisfied  by  the  co-ordinates  of  all  points  of  the  instantaneous 
axis  in  all  of  its  positions.  It  is  therefore  the  equation  of  the 
rolling  cone. 

By  replacing  A,  B,   C  within   the  parentheses  by  -5,  j,,  -5, 

respectively,  a^  b^  c  being  the  semi-axes  of  the  ellipsoid,  the 
equation  of  the  rolling  cone  becomes 

Aj  -  i\'' + ^(7  -  -^y + <7  -  7>' = °-  (^7-^) 

The  position  and  character  of  this  cone  depend  on  the  values 
of  the  constant/. 

110.  Discussion  of  the  Rolling  Cone. 
(i)  Let/  =  a.     Eq.  (274)  becomes 


ROTATION  DUE    TO  IMPULSIVE  FORCES.  121 

which  is  satisfied  only  by 

jt:  =  —      y  =  o     and     z  -=■  o, 

o 

Hence  the  axis  of  x  is  the  rolling  cone,  the  directing  cone,  the 
invariable  axis,  the  instantaneous  axis,  and  the  longest  principal 
axis  of  the  body.     The  body  rotates  uniformly  about  this  axis. 

(2)  Let  ay  py  b.  The  second  and  third  terms  of  Eq.  (274) 
then  have  the  same  sign;  x  is  then  the  axis  of  the  cone,  and  the 
sections  of  the  cone  normal  to  x  are  ellipses. 

(3)  Let/  =  b.     Eq.  (274)  becomes 

and  we  have 

which  are  the  equations  of  two  planes  equally  inclined  to  the 
principal  plane  ab  of  the  ellipsoid  and  intersecting  in  the  mean 
principal  axis.  They  cut  from  the  ellipsoid  two  equal  ellipses, 
which  are  called  the  critical  ellipses,  or  separating polhodes,  of  the 
central  ellipsoid.     The  semi-axes  of  the  critical  ellipses  are 


sT. 


a  c 


a^  -\-c'' TT       and     ^ (278) 


(4)  Let  by- py  c.  Then  the  first  and  second  terms  of  Eq. 
(274)  have  the  same  sign;  z  is  then  the  axis  of  the  cone,  and  the 
sections  of  the  cone  normal  to  z  are  ellipses. 

(5)  Let/  =  c.     Then 


o 


122 


MECHANICS   OF  SOLIDS. 


and  the  axis  of  z  is  the  rolling  cone,  the  directing  cone,  the  in« 
variable  axis,  the  instantaneous  axis,  and  the  shortest  principal 
axis  of  the  body.     The  bod}-  rotates  uniformly  about  this  axis. 

III.  The  Polhode  and  Herpolhode. — In  the  ist  and  5th  cases 
the  polhode  and  herpolhode  are  points. 

In  the  2d  and  4th  cases  the  polhode  is  in  general  a  curve  of 


Fig.  44. 


double  curvature,  and  the  herpolhode  is  a  wavy  curve,  as  indi- 
cated in  Fig.  44. 

In  Fig.  44  let  «  >/  >  b,  take  the  tangent  plane  to  be  the 
horizontal  plane,  and  assume  the  vertical  plane  parallel  to  the 
plane  of  the  longest  and  shortest  principal  axes.  Then  the  long- 
est, shortest  and  mean  axes  are  projected  equal  to  themselves 
in  aa'\  CC  and  bb\  respectively;  //'/"  is  the  vertical  projection 


ROTATION  DUE    TO  IMPULSIVE  FORCES.  1 23 

of  the  polhode,  and  hh'h"h"\  etc.,  is  the  herpolhode;  the  invari- 
able axis  is  projected  in  00"  and  C?',  and  the  instantaneous  axis 
in  Op  and  O'h',  OE'  and  (9^  are  traces  of  the  cyclic  planes,  and 
OE"  and  OE"*  are  traces  of  the  planes  of  the  critical  ellipses. 

The  maxima  and  minima  values  of  d  recur  when  the  vertices 
of  the  polhode  come  into  the  tangent  plane.  These  vertices  are 
the  intersections  of  the  polhode  by  the  principal  planes  of  the 
ellipsoid.  The  maxima  and  minima  values  of  the  radius  vector  of 
the  herpolhode  correspond  to  those  of  d,  and  hence  this  curve 
will  lie  between  the  circumferences  of  two  circles  whose  common 
centre  is  (9',  and  whose  radii  are  O'h  and  0'h\  corresponding  to 
the  greatest  and  least  values  of  S.  If  the  angle  included  between 
two  consecutive  maximum  radii-vectores  of  the  herpolhode  be 
commensurable  with  a  right  angle,  the  curve  will  be  retraced 
after  a  certain  number  of  complete  turns,  and  the  herpolhode 
will  be  a  closed  curve.  If  this  angle  be  incommensurable  with 
a  right  angle,  the  instantaneous  pole  will  never  retrace  its  former 
path  on  the  tangent  plane. 

The  general  value  of  the  radius  vector  of  the  herpolhode  is 
given  by  the  equation 


p"  =  <s'-^  =  *'-/; (279) 

from  which,  together  with  ds  =  ds\  in  which  ds  is  the  length  of 
its  elementary  arc,  while  ds*  is  that  of  the  corresponding  element 
of  the  polhode,  the  curve  may  be  found. 

In  the  3d  case  the  herpolhode  becomes  a  spiral  whose  pole  is 
O'  and  whose  maximum  radius  vector  is 


corresponding  to  an  instantaneous  axis  coincident  with  the  semi- 
transverse  axis  of  the  critical  ellipse. 


124  MECHANICS  OF  SOLIDS. 

A  complete  analysis  shows  that  if  the  rotation  be  about  b  it 
will  remain  so,  but  if  the  body  be  started  to  rotate  about  any 
other  diameter  of  the  critical  ellipse  it  will  require  an  infinite 
time  for  the  instantaneous  axis  to  reach  b. 

112.  Permanent  Axes. — To  explain  what  is  understood  by  per- 
matient  axes  of  rotation,  let  the  initial  impulsion  be  such  as  to 
cause  the  body  to  rotate  about  any  of  the  central  principal  axes. 
Then,  as  we  have  seen,  the  polhode  and  herpolhode  are  coincident 
points,  and  the  instantaneous,  principal  and  invariable  axes  are 
initially  coincident,  and  will  continue  so  during  the  entire  mo- 
tion. For  this  reason  the  three  central  principal  axes  are  called 
permanent  axes  of  rotation;  the  discussion  shows  that  they  are 
the  only  lines  which  have  this  property  of  permanence. 

113.  Stability  of  Rotation. — The  critical  ellipses  divide  the  sur- 
face of  the  ellipsoid  into  four  areas,  two  surrounding  the  ex- 
tremities of  the  shortest  axis,  and  the  other  two  the  extremities 
of  the  longest  axis.  Within  these  areas  the  corresponding  pol- 
hodes  appertaining  to  each  axis  are  found. 

If  the  impulsion  be  such  as  to  develop  initially  an  instanta- 
neous axis  very  near  the  shortest  axis  of  the  ellipsoid,  the  corre- 
sponding polhode  will  be  a  small  curve  surrounding  its  extrem- 
ity, and  the  successive  positions  of  the  instantaneous  axis  meeting 
the  surface  in  this  polhode  will  never  depart  very  far  from  the 
shortest  axis.  It  will  periodically  return  to  its  initial  position  in 
the  body,  after  passing  through  its  two  maximum  and  minimum 
displacements  with  respect  to  the  shortest  axis.  This  will  like- 
wise be  true  for  any  initial  or  subsequent  additional  impulsion 
which  causes  the  polhode  to  lie  within  the  assumed  area.  If  the 
initial  impulsion  develops  an  instantaneous  axis  whose  polhode 
surrounds  the  longest  axis,  the  successive  instantaneous  axes 
will  be  related  to  that  axis  in  a  precisely  similar  manner.  But 
since  the  mean  principal  axis  lies  in  the  planes  of  the  critical 
ellipses,  any  instantaneous  axis  not  in  one  of  these  planes,  how- 
ever near  it  may  be  to  the  mean  axis,  will  belong  to  a  polhode 
surrounding  the  longest  or  shortest  axis,  and  will  depart  far 
from  the  mean  axis.     The  longest  and  shortest  axes  are  there- 


ROTATION  DUE    TO  INCESSANT  FORCES.  12$ 

fore  called  stable  axes  of  rotation,  and  the  mean  axis  an  unstable 
axis. 

When  a:=  b  the  central  ellipsoid  is  an  oblate  spheroid,  and 
the  critical  ellipses  unite  in  the  equator.  The  areas  reduce  to 
two,  surrounding  the  axis  of  the  spheroid,  which  is  the  only- 
stable  axis  of  rotation,  and  the  polhodes  are  circles  whose  pole 
is  the  extremity  of  the  axis. 

When  b  ^=  c  the  central  ellipsoid  becomes  a  prolate  spheroid, 
the  critical  ellipses  unite  in  its  equator,  and  the  axis  of  the 
splieroid  is  the  only  axis  of  stability.  An  elongated  rifled  pro- 
jectile is  such  a  body,  and  its  axis  is  the  only  stable  axis  of  rota- 
tion. A  very  great  initial  angular  velocity  is  usually  given  to  it 
about  this  axis,  so  that  the  influences  which  modify  its  angular 
velocity,  either  as  to  amount  or  change  of  axis,  while  the  pro- 
jectile is  describing  its  trajectory,  will  be  comparatively  so  mi- 
nute as  not  to  cause  the  instantaneous  axis  to  depart  sensibly 
from  this  axis  of  stability. 


Rotation  Due  to  Incessant  Forces. 

114.  Euler's  Equations  of  Rotation. — Differentiating  Eqs.  (251), 
we  have 


d^y'       fdoOz       JgOjc  ,       ,  ,  ,     .  ,  ,  ,      . 

-J^=X'  --^-Z'-J^^QOJ^Z'  ODy-y'  QO,)-CO^{y  QO^-X'  ODyY 

d^z'       JoSjc       ,doOy  ,         ,  ,  ,      ^  ,  - 


.  (281) 


and,  substituting  in  Eqs.  (Rm),  the  latter  become,  after  omitting 
accents, 


126 


MECHANICS  OF  SOLIDS. 


—  ^mxy(pOx  —  ^^^  —  '^^^yA~~j7'  "f~  ^x^z\ 


^  idCD:c 


_^     zd'^X  —  xd'^Z  ^,21       3\^'«'y     .      -sr-     /  a  a\ 

—  '2mxz(csO^—  QDx)  —  ^mxyi—r-  +  ODyCoA 

—  :Smyz  [-^  —  GOxGJj]  =  M; 

—  2myz(Qo/  —  G?/)  —  '2.mxz\--Tj-  +  c^a.ca?' j 

—  2mxy{--^  —  GOxGoA  =  N, 


(282) 


Equations  Rm  and  (281)  are  true  whatever  be  the  directions 
of  the  axes  x^,  y\  2'.  Let  the  co-ordinate  axes  be  the  principal 
axes  of  the  body;  then  Eqs.  (282)  may  be  written 


B-^^{C-A)oOxOD,  =  M; 

A-^-{B-C)GDyGO,  =  N. 


.     (283) 


These  equations  are  known  as  Euler's  equations  of  rotation, 
and  the  values  of  g?^,  ooy  and  oOz  may  be  found  from  them  when 
integration  is  possible.     But  since  the  principal  axes  conform  to 


ROTATION  DUE    TO  INCESSANT  FORCES. 


127 


■the  motion  of  the  body,  the  complete  solution  requires  that  the 
position  of  these  axes  at  any  instant  shall  be  determined  with 
respect  to  the  axes  fixed  in  direction. 

These  equations,  like  Eqs.  (Rm),  also  apply  to  rotation  about 
a  fixed  axis  or  a  fixed  point. 

115.  Auxiliary  Angles. — Let  X^  V,  Z,  Fig.  45,  be  the  axes  fixed 
in  direction,  and  X\  Y%  Z'  the  principal  axes  of  the  body. 
Conceive  a  sphere  to  be  described  Z_ 

about  O  with  unit  radius.  Its 
intersections  with  the  co-ordinate 
planes  are  xy,  xz^  yz,  x'y\  x'z\ 
y'z\  and  the  intersection  of  the 
planes  X  V  and  X'  Y'  is  ON.  As- 
sume the  notation 


X'NY=ZOZ'  =  6; 
XON  =^; 
X'ON  =  0. 


{284) 


VJV  is  called  the  line  of  the  nodes, 

6  the  obliquity,  and  tp  thQ  precession.  ^'°"  '*^* 

Taking  Z'  positive  when  Z'OZ  <  90°,  we  have  from  the  spherical 

triangles  of  the  figure,  considering  ^  as  a  vertex  in  each, 


cos  xOx'  =  cos  0  cos  rj)  —  sin  0  sin  tj)  cos  ^; 

cos  xOy'  =  —  sin  0  cos  ip  —  cos  0  sin  tp  cos  d\ 

cos  xOz^  =  sin  tp  sin  d; 

cos  yOx'  =  cos  0  sin  ^  -f-  sin  0  cos  ip  cos  6; 

cos  yOy^  =  —  sin  0  sin  ip  -f-  cos  0  cos  tp  cos  6; 

cos  yOz'  =  —  cos  tp  sin  6; 

cos  zOx'  =  sin  0  sin  6; 

cos  zOy'  =  cos  0sin  6; 

cos  zOz'  =  cos  ^. 


(28s) 


116.  The  component  angular  velocities  about  ON,  Zand  Z' 
are  frequently  used  as  auxiliary  angular  velocities  for  the  deter- 


128 


MECHANICS  OF  SOLIDS. 


.      ,     dd  dip       ^dd) 
mination  of  oox,  oo,  and  g?^;  they  are  respectively  — -,  -j~  and  -~. 

The  first  is  called  the  nutation  of  obliquity^  and  the  second  the 
precessional  velocity.  The  latter  is  direct  when  ^  increases  with 
the  time,  and  retrograde  when  it  decreases. 

For  the  angles  which  the  axes  Z,  Z'  and  the  line  ON  make 
with  the  axes  X\  Y\  Z'  we  have 


cos  ZOX*  =  sin  <p  sin  6;  "| 
cos  ZO  y  =  cos  0  sin  6; 
cos  ZOZ^  =  cos  6; 

cos  Z'OX'=  o; 
cos  Z'OV'=  o; 
cos  Z'OZ  =  i; 

cos  NOX'  =  cos  0; 
cos  iV^(9y=  —  sin  0; 
cos  iV^6>Z'  =  o; 


(286) 


and  hence,  by  the  principle  of  the  parallelopipedon  of  angular 
velocities,  we  have 


C0x=       ~77^^^^  '^  dt  ^^"  ^  ^^"    ' 

de     .      ^    ,    dip  .     .     n 

Wy  =  —  — -  sin  0  +  -^  cos  0  sin  C^; 


.  (287) 


117.  T/te  Gyroscope, — The  problem  of  the  gyroscope  illustrates 
this  subject.     It  may  be  stated  thus: 

Find  the  circumstances  of  motion  of  a  solid  of  revolution' 
about  a  fixed  point  on  its  axis,  it  having  been  given  an  initial 


ROTATION  DUE    TO  INCESSANT  FORCES. 


129 


rotation  about  its  axis  and  then  left  to  the  action  of  its  own 
weight. 

Let  (9,  Fig.  46,  be  the  fixed  point, 
Ox^  Oy,  Oz  the  fixed  axes,  and  Ox',  Oy\ 
Oz'  the  principal  axes  at  O,  Oz  being 
taken  vertical  and  positive  upward.  Oz' 
is  the  axis  of  revolution  of  the  body. 
Let  h  be  the  distance  from  O  to  the 
centre  of  gravity,  and  assume  A  =■  B 
and  B  <C. 

The  components  of  the  weight  in  the 
directions  of  Ox',  Oy'  and  Oz'  are 


Fig.  46. 


X'  =  mg  cos  (180°  —  zOx')  =z  —  mg  cos  zOx'-, 
y  =  mg  cos  (180°  —  zOy')  =  —  mg  cos  zOy'\ 
Z'  =  mg  cos  (180°  —  zOz')  =z  —  mg  cos  zOz'\ 


(288) 


and  the  component  moments  of  the  weight  with  respect  to  the 
same  axes  are,  Eqs.  (286), 

N  =Z'y'  —  Y'z'  =      mgh  cos  zOy'  =      vigh  sin  ^  cos  0;  ) 
M=X'z'—  Z'x'  =  —  mgh  cos  zOx'=—  mgh  sin  0  sin  0;  >  (289) 
Z  =  Y'x'-  X'y'=o.  ) 


Substituting  these  values  in  Euler's  Eqs.  (283),  we  have 


dt 

dOOy 
~dt 


—  {A  —  C)GOyGDz  =  iV=      mgh  sin  6  cos  0; 
-\-  {A  —  C)gDxOOz  =  M  =  —  mgh  sin  6  sin  0; 


^(290) 


Integrating  the  last  equation,  we  have 

Cl»2  =  a  constant  =■  n (291) 


130  MECHANICS   OF  SOLIDS. 

Multiplying  the  first  of  Eqs.  (290)  by  (y^and  the  second  by 
GOy^  we  have  by  addition,  Eqs.  (287), 


A\QOx-Tr  +  00y—zj\  =  mgh  sm  uyoo^  cos  0  --  CsJ^sin  0) 


=  mgh  sin  c'— . 


(292) 


Integrating  this  equation,  we  have 

iA{GoJ  +  gd/)  =  ^«^y^  (cos  ^0  —  cos  6),      ,    .     (293) 
in  which  6^  is  the  initial  angle  zOz\ 

Adding  the  initial  kinetic  energy  of  rotation  iCn^,  we  have 
iAiGoJ"  +  gd/)  +  iCfi"  =  mgh  (cos  B^  -  cos  6)  +  iC•«^  (294) 

Since  mgh  (cos  ^^  —  cos  6)  is  the  work  of  the  weight  while  G 
falls  over  the  distance  h  (cos  Q^  —  cos  ^),  we  see  that  Eq.  (294) 
expresses  the  theorem  of  kinetic  energy  of  rotation. 

118.  From  Eqs.  (287)  we  readily  get 

<»/  +  <»/  =  5 +  sin»^fl;.    .    .    .    (295) 

which  substituted  in  Eq.  (294)  gives 

A-j-^  +  A  sin''  Q-j-i,  —  2mgh{cos  6^  —  cos  6).     .     (296) 
at  at 

Multiplying  the  first  and  second  of  Eqs.  (290)  by  sin  0  and 
cos  0  respectively,  we  have,  by  addition, 


ROTATION  DUE    TO  INCESSANT  FORCES  I3I 

^[sin  0—^+cos0-^j+«(C— -(4) [cousin  0—0?;^ cos  0]  =0,(297) 

and,  since  Eqs.  (287)  give 

ooy  sin  0  —  fl?^  cos  0  =  —  ^,  .    .    .    ,    (298) 
Eq.  (297)  reduces  to 

^^sin  0-^  +  cos  0-^j  +  nA-j^  -  «C-^  =  o.      (299) 
We  have  also,  from  Eqs.  (297), 


cOx  sin  0  +  (Wy  cos  0  =  sin  ^-^.     .     ,     .     (300) 


Differentiating  this  last,  dividing  by  dt  and  reducing  by  the 
relations  of  Eqs.  (287),  we  have 

sin  0^^  +COS  0-^  =  cos  ^  ^  +  sin  ^ 


{gOx  cos  0  —  G7y  sin  0) 


^0 


Substituting  this  value  in  Eq.  (299),  we  have 

./  jedil,\     .     uPtp\       ^dd  ,      . 


132 


MECHANICS  OF  SOLIDS, 


which,  after  multiplying  by  sin  d  and  integrating,  gives 


A  sin'  QJ-  +  CV/  cos  ^  =  a  constant  =  ^';  .     .     (303) 


or  between  limits. 


A  sin'  e^^  =  C«(cos  B^  -  cos  6), 


(304) 


From  the  last  of  Eqs.  (287)  we  also  have,  since  aOg  =  n. 


dcf)      dtp 


(305) 


Eqs.  (296),  (304)  and  (305),  viz., 


A-—^  +  A  sin'  6-^^  =  2mgh  (cos  Q^  —  cos  6), 


A  sin'  0-^  =  Cn  (cos  d^  —  cos  ^), 


d(f)       dip         ^ 


(306) 


are  the  differential  equations  of  motion  of  the  gyroscope.     From 

them  the  values  of  6,  tp  and  <p,  which  give  the  position  of  the 

body  at  any  instant,  may  be  found  in  terms  of  known  quantities. 

119.  Square  the  second  of  Eqs.  (306)  and  multiply  the  first 

by  A  sin'  6  and  eliminate  -—-  from  the  resulting  equations  by 
subtraction;  then  solve  with  respect  to  — -,  and  we  have,  for  the 


ROTATION  DUE    TO  INCESSANT  FORCES.  1 33 

nutational  velocity, 

de_  .  (cos  ^,-cos  ey^  \2mghA  sin'  e-C^n^  (cos  ^,-cos  6)1^ 


Any  value  of  6  less  than  B^  makes  —  imaginary,  and  there- 
fore the  centre  of  gravity  can  never  get  above  its  initial  position. 
And  since  —  is  imaginary  when  6'  =  180°,  the  centre  of  gravity 
can  never  reach  the  vertical  through  the  fixed  point. 

Making  —  =  o  in  Eq.  (307),  we  find  that  the  resulting  equa- 
tion may  be  satisfied  by  6^  =  ^„,  and  also  by  the  two  roots  of  the 
equation 

2mghA  sin'  d  —  C"«'(cos  6^  —  cos  6)  =  o.  .     .     (308) 


One  of  these  roots  gives  a  maximum  value  of  6,  which  we  will 
call  ^,.  The  other  root  gives  cos  0  >  i,  and  hence  corresponds 
to  no  angle.  Thus  we  see  that  the  body  falls  from  its  initial 
position  until  6  =  6^,  then  rises  until  6  =  6^,  and  continues  to 
oscillate  between  these  two  values.  The  integral  of  dt  (Eq.  307) 
between  the  limits  0^  and  6^  evidently  gives  half  the  time  of  a 
complete  nutational  oscillation. 

The  precessional  velocity  given  by  the  second  of  Eqs.  (306)  is 


d(p       Cn{cos  6^  —  cos  6)  ,      . 

-di  = ASTxTe : "°9) 


and  it  is  zero  when  6  =  6^,  and  a  maximum  when  6  =  6^.     It  is 
direct  or  retrograde  according  as  n  is  positive  or  negative. 

Combining  the  motions  in  nutation  and  precession,  we  find 
that  the  horizontal  projection  of  the  path  of  the  centre  of  gravity 


134  MECHANICS  OF  SOLIDS. 

lies   between    two   concentric    circumferences   whose    radii    are 
h  sin  d^  and  h  sin  ^„  and  is  tangent  to  the  outer  and  normal  to 
the  inner  circumference. 
From  Eq.  (308)  we  have 


•     a       r-    i/cos^.  —  cos^  .      . 

s.n  e  =  C«  f  — ^-p^^— (3,0) 


from  which  we  see  that  B^  —  6^  may,  by  increasing  the  value  of 
«,  be  made  less  than  any  assignable  quantity.  In  the  common 
gyroscope  we  can  give  n  such  a  value  that  the  eye  can  detect 
neither  the  vertical  motion  nor  the  variation  in  the  precessional 
velocity. 

This  discussion  gives  only  the  general  character  of  the  mo- 
tion. The  complete  solution  of  the  problem  requires  the  inteorra- 
tion  of  Eqs.  {306),  and  this  involves  methods  not  given  in  the 
course  of  mathematics  at  the  Military  Academy. 


Impact. 

120.  When  two  bodies  collide  there  is  a  transfer  of  energy 
from  one  to  the  other,  by  which  changes  in  the  velocities  of  both 
bodies,  and  in  their  form  and  volume,  are  effected.  The  impact, 
though  ordinarily  said  to  be  instantaneous,  requires  a  finite  time 
for  its  completion.  When  after  collision  we  examine  the  sur- 
faces of  two  ivory  balls  which  have  been  previously  oiled,  we 
notice  that  their  areas  of  contact  during  collision  must  have 
been  very  much  greater  than  when  they  simply  rest  against  each 
other.  Hence  the  distance  between  their  centres  of  mass  during 
impact  must  be  less  than  the  sum  of  the  radii  of  the  spheres,  and 
the  intensity  of  the  mutual  pressure  of  the  colliding  bodies  evi- 
dently varies  by  continuity  from  zero  to  a  maximum  and  then  to 
zero  again.  The  instant  of  nearest  approach  of  their  centres 
separates  the  period  of  compression  from  that  of  restitution^  and  at 
that  instant  their  centres  have  the  same  velocity. 


IMPACT.  135 


Actual  solids  possess  a  certain  degree  of  elasticity  of  form 
and  volume,  by  which  they  regain  approximately  their  original 
form  and  volume  when  the  impact  has  ended.  If  C  be  the  in- 
tensity of  the  impulsion  producing  compression,  R  that  which 
restores  the  form  and  volume,  and  e  their  ratio,  we  have 

R^eC', (311) 

e  is  called  the  coefficient  of  restitution^  and  in  actual  bodies  is  less 
than  unity  and  greater  than  zero. 

121.  Direct  and  Central  Impact. — Let  a  spherical  mass  w,  mov- 
ing with  a  velocity  z;,  collide  with  a  similar  mass  m' ,  whose  ve- 
locity is  u  in  the  same  direction,  and  let  w  be  their  common 
velocity  at  the  instant  of  nearest  approach  of  their  centres;  then, 
taking  velocities  in  opposite  directions  to  have  opposite  signs, 
we  have 

C=  tn{v  —  w)  =  m'{w  —  «),      ....     (312) 


whence 


and 


C= — i — }{v—u) (313) 


i?  =  a=-^^(z;-«y (314) 


Let  Fand  C/'be  the  final  velocities  of  m  and  m\  and  we  have 

je  +  c  =  c(i  + .)  =  ^^^(t,  -  «)(i  + .) 

=  m{v-V)=m\U-u) (315) 

I^nce 


136 


MECHANICS   OF  SOLIDS. 


V=V 


m' 


m-\-  m 


-,(v-  «)(i  +^); 


U:=zu-\- 


-,(v  -  u)(T^e\ 


(316) 


7?i  -j-  nv 
The  limits  of  these  values,  found  by  making  ^  =  i  and  e  —  o, 


are 


and 


2m' 
V  =  v — 


-,{^  -  ^)f 


U^=u-{- 


2m 


m-\-m 


li^  -  «)» 


y  =  y [p   ^U)=  ; J-, 


"  m-\-m 


-(z;  -  «)  = 


m  -\-  m'     ^ 


(317) 


(318) 


In  actual  cases  ^  is  a  constant  to  be  determined  by  experi- 
ment. Whatever  its  value  may  be,  it  is  readily  seen  from  Eqs. 
(316)  that  the  sum  of  the  momenta  of  the  two  bodies  after  colli- 
sion is  equal  to  that  before  collision;  therefore  no  momentum  is 
destroyed  by  the  impact. 

The  sum  of  the  kinetic  energies  of  the  masses  before  and 
after  impact  are  respectively 


i(wz/'4-wV) (319) 


and 


mm 


i(mi^  +  m'u')-i-^^-^iv-uY(i-e').    .    .     (320) 


Whence  we  see  that  a  loss  of  kinetic  energy  always  accompanies 
the  impact  of  actual  masses. 


IMPACT. 


137 


122.  Oblique  Impact. — If  the  paths  of  the  colliding  bodies  be 
oblique  to  each  other,  their  velocities  may  be  resolved  into  com- 
ponents in  the  directions  of  the  common  tangent  and  normal  at 
the  point  of  impact.  C  and  R  will  depend  only  on  the  normal 
components,  the  tangential  components  having  no  effect  on  the 
impact.  The  final  velocity  of  each  body  will  therefore  be  given 
by  the  resultant  of  the  changed  normal  component  and  the  un- 
changed tangential  component. 

123.  If  m'  be  very  great  with   respect   to  m  and   at  rest  we 
have  the  case  of  impact  against  a  fixed  obstacle.    -; 
Then  if  0  be  the  angle  of  incidence,  Fig.  47,  or  that      |  — 
which   the  direction  of  the  path  of  m  makes  with 
the  normal  to  the  deviating  surface  at  the  point  oifn'l 
impact,  we  have  for  the  component  impulsions  of 
compression 


mv  sin  (p        and        mv  cos  0, 


Fig.  47. 


and  for  those  of  restitution 


mv  sin  0        and         —  mev  cos  0; 


the  resultant  of  the  latter  being 


mv  4^sin'0H-^'  cos'0. 


If  the  angle  of  reflection  is  0'  we  have 

sin  0  tan  0 


tan  0'  = 


e  cos  0 


(3") 


which  varies  between  —  tan  0for  ^  =  i,  and  00  for  ^  =  o.  Hence 
in  all  actual  cases  of  impact  the  path  of  the  reflected  body  will 
make  an  angle  with  the  normal  greater  than  that  of  incidence 
and  less  than  90°,  depending  on  the  value  of  e  for  the  bodies 
considered. 


138 


MECHANICS  OF  SOLIDS. 


Axis  of  Spontaneous  Rotation. 

124.  A  spontaneous  axis  is  a  right  line  fixed  in  space^  about 
which  a  free  body  rotates  during  impact  while  the  centre  of 
mass  of  the  body  is  in  motion.  Its  position  and  the  necessary 
conditions  for  its  development  are  derived  from  Eqs.  (117). 
Dividing  these  by  dt,  and  replacing  the  component  angular 
velocities  about  the  centre  of  mass  by  their  symbols,  we  have 


dx      dx.    ,      -  - 

dz      dz  .  , 


(322) 


in  which  the  first  members  are  the  component  velocities  of  any 
molecule  of  the  body  with  reference  to  a  set  of  axes  fixed  in 
space,  and  the  first  terms  of  the  second  member  are  the  compo- 
nent velocities  of  the  centre  of  mass  with  respect  to  the  fixed 
origin,  when  the  centre  of  mass  is  taken  as  the  movable  origin. 
From  Eqs.  (Tm')  we  have 


'dt~^''~  M' 

dt~    ''~  M' 

^-v  -  ^. 

dt  ~    '~  M' 


'     (323) 


in  which  X^,  Y^,  Z^  are  the  component  intensities  of  the  result- 
ant impulsion  in  the  direction  of  the  co-ordinate  axes. 


AXIS   OF  SPONTANEOUS  ROTATION. 


139 


Substituting  these  values  in  Eqs.  (322),  we  have 


dt       M  ^ 


y  oD^ 


dt       M 


+  x'ooz  —  z'0Ojc\ 


dz        Z.    .      .  , 


(324) 


For  all  points  at  rest  with  respect  to  the  fixed  origin  we  have 
the  conditions 


dx  _  dy  _dz  _ 
~dt~~dt~~dt~^' 


(325) 


and  Eqs.  (324)  become 


M 


*--\-  z'ooy  —y'GO^  =  o; 


(326) 


which  will  be  the  equations  of  a  right  line  fixed  in  space  when 

X,(w^  +  K,(»y  +  Z.tt?,  =  o (327) 

Dividing  Eq.  (327)  by  R,qo,  we  have 


R.    oo~^  R.    00^  R.   GO         * 


(328) 


which  expresses  the  condition  that  the  action-line  of  the  result- 
ant impulsion  is  perpendicular  to  the  instantaneous  axis;  hence 


I40  MECHANICS  OF  SOLIDS. 

we  conclude  that  a  spontaneous  axis  will  be  developed  when  a  body  is 
so  struck  as  to  make  the  instantaneous  axis  perpendicular  to  the  result- 
ant impulsion;  otherwise  Eqs.  (326)  are  the  equations  of  a  single 
point  whicli  alone  is  at  rest  for  the  instant.  We  see  also  by 
comparing  Eqs.  (326)  with  Eqs.  (252)  that  the  spontaneous  axis  is 
always  parallel  to  the  instantaneous  axis. 

125.  It  is  readily  seen  from  Eq.  (327)  that  the  required  con- 
dition can  be  satisfied  only  when  at  least  one  factor  of  each  term 
is  zero;  that  is,  either  when  the  line  of  impact  lies  in  a  central  prin- 
dpal  plane  or  when  it  is  parallel  to  a  central  principal  axis.  The  dis- 
cussion is  the  same  for  all  cases. 

126.  Let  the  line  of  resultant  impact,  Fig.  48,  be  parallel  to 
the  principal  axis  y'  and  lie  in  the  prin- 
cipal plane  x'y',  and  let  R^  =  J/Fbe  the 
intensity  of  the  resultant  impulsion;  then 
we  have 

X,  =  o;     Y^  =  MV-  Z,  =  o; 

MVh     )-  (329) 


GOr  =  0:     G?v  =  O;  ODz  = 


Fig.  48. 


h  being  the  lever  arm  of  the  impulsion 
with  respect  to  the  axis  z'\  Eq.  (327)  is  satisfied,  and  the  equa- 
tions of  the  spontaneous  axis  developed  by  the  impact  are 

,       o         ,  C  Mk;  k;      .       , 

which  are  those  of  a  right  line  parallel  to  the  principal  axis  z\ 

intersecting  the  axis  x'  at  a  distance  from  the  centre  of  mass 

k  ' 
■equal  to j-. 

Let  /  be  the  perpendicular  distance  between  the  line  of  im- 
pact and  the  spontaneous  axis;  then  we  have 

/  =  '5  +  ^; (331) 


AXIS  OF  SPONTANEOUS  ROTATION.  I4I 

whence 

{i-K)h  =  k: (332) 

Since  /&/,  the  square  of  the  principal  radius  of  gyration  of 
the  body  with  respect  to  the  axis  2',  is  constant,  we  see  that  the 
two  distances,  viz.,  //  from  the  centre  of  mass  to  the  line  of  im- 
pact, and  I  —  h  from  the  centre  of  mass  to  the  spontaneous  axis,, 
are  reciprocally  proportional;  hence,  as  the  line  of  impact  re- 
cedes from  the  centre  of  mass,  the  spontaneous  axis  approaches 
that  point,  and  conversely.  When  the  line  of  impact  passes 
through  the  centre  of  mass  the  spontaneous  axis  is  at  an  infinite 
distance,  as  it  should  be,  since  in  this  case  the  body  will  have 
motion  of  translation  only. 

127.  When  a  spontaneous  axis  is  developed  the  action-line  of 
the  corresponding  impulsion  is  called  an  axis  of  percussion^  and 
each  of  its  points  a  centre  of  percussion;  the  latter  term,  however,, 
being  generally  applied  to  the  point  in  which  the  axis  of  per- 
cussion intersects  the  line  h.  The  corresponding  point  of  the 
spontaneous  axis  is  called  a  centre  of  spontaneous  rotation.  The 
axis  of  percussion  and  the  spontaneous  axis  are  conjugate  lines, 
each  of  which  implies  the  other;  the  positions  of  both  are  con- 
nected and  determined  by  Eq.  (332).  For  example,  the  spon- 
taneous axis  of  a  straight  rod  struck  at  its  extremity  in  a  direc- 
tion perpendicular  to  its  length  is,  Eq.  (222),  given  by 

(/-/i)yi=  (/-«)«  =  !'; (333). 

whence 

/  —  flj  =  J^,  or    /  =  |. 2«, (334). 

or  is  at  a  distance  two  thirds  of  the  length  of  the  rod  from  the 
line  of  impact.  All  elements  of  the  rod  beyond  the  spontaneous 
axis  will  have  a  motion  in  a  direction  opposite  to  that  of  the 
impact,  and  those  between  the  axis  and  line  of  impact  a  motion 


142  MECHANICS   OF  SOLIDS. 

in  the  same  direction  as  the  impact.  This  explains  the  cause  of 
the  physical  shock  experienced  when,  in  striking  a  ball  with  a 
bat  or  in  chopping  with  an  axe,  the  part  held  by  the  hand  does 
not  conform  to  the  position  of  the  spontaneous  axis  correspond- 
ing to  the  line  of  impact. 

128.  In  Eq.  (332)  substitute  I  —  h  ior  h,  and  we  have 

[i-{i-h)-\(i-h)  =  h{i-k)  =  k;. .   .   .   (335) 

Hence,  for  parallel  impacts,  the  centre  of  percussion  and  the 
centre  of  spontaneous  rotation  are  reciprocal  and  convertible; 
that  is,  if  the  centre  of  spontaneous  rotation  become  a  new  cen- 
tre of  percussion,  the  old  centre  of  percussion  will  become  the 
new  centre  of  spontaneous  rotation. 


Constrained  Motion. 

129.  When  a  rigid  surface  or  curve  deflects  a  body  from  the 
free  path  which  any  given  system  of  forces  would  cause  it  to 
take,  the  motion  is  said  to  be  constrained.  Let  the  motion  of  its 
centre  of  mass  determine  the  translation  of  the  body,  and,  by  the 
principles  in  Art.  82,  we  may  omit  the  present  consideration  of 
its  motion  of  rotation  about  that  point.     Eq.  (119)  then  becomes 

(^-^S"'-^ + ( ^-^S)'^^ + (^-^S)'^^ = °-  (336) 

130.  Equations  of  Constraint. — Let 

L^f(x,y,z)  =  o (337) 

be  the  equation  of  the  surface  upon  which  the  centre  of  mass  is 
constrained  to  move.     Differentiating  this  equation,  we  have 

g^^  +  ^^_,  +  ^^,  =  o (338) 


CONS  TRA I  NED   MO  TION. 


143 


Since  the  path  of- the  centre  of  mass  lies  in  the  given  surface, 
Eqs.  {zz^)  ^"d  {ZZ^)  maybe  combined  by  considering  only  those 
values  of  dXy  dy  and  dz  which  are  common  to  the  path  and  sur- 
face. To  make  the  terms  of  these  equations  quantities  of  the 
same  kind,  multiply  the  differential  equation  of  the  surface  by 
an  intensity  /.  Add  the  resulting  equation  to  Eq.  (336),  and  we 
have 

('-<i+'S}-+(>---S:+'S)* 

Now,  if 

.dL      ^dL          .      .dL  .      ^ 

^^'    ^^     ""^     ^^' (340) 


be  the  rectangular  components  of  a  force  which,  together  with 
the  given  extraneous  forces,  will  cause  the  body  to  remain  con- 
tinually on  the  geometrical  surface  whose  equation  is  that  of  the 
rigid  surface  (337),  the  latter  may  be  supposed  removed  and  the 
body  will  be  a  free  body  subjected  to  the  action  of  the  com- 
ponent extraneous  forces  X -{- 1-^,  etc.;  hence,  by  Eqs.  (Tn,),  we 
will  have 


dx 


dL         .^d^x 


dz 


dt 


•        •        • 


(341) 


Eliminating  /  from  these  equations,  we  have 


144 


MECHANICS  OF  SOLIDS. 


(.- 


M 


dy 
•\dL  _ 


dt^'Jdz 


dt'ldx  ~ 


>. 


(342) 


which  are  called  the  differential  Equations  of  Constraint^  and  from 
which,  with  the  equation  of  the  surface,  the  path  of  the  centre 
of  mass  and  its  position  at  any  time  may  be  determined. 

131.   The  Normal  Reaction. — Let  N  represent  the  intensity  of 
the  resultant  whose  component  forces  are 


-dL         dL  dJL 

dx^        dy  dz^ 


and  Qjc->  ^y^  ^z  the  angles  which  N  makes  with  the  co-ordinate 
axes;  then 


„         I  dL 

cos  d:c  =  -T-T-J-  = 

N  dx 


dL 

dx 


cos     dy 


^  dx"  '^  df'^  dz" 
dL 
L  dL  dy 


"T  ^,.3  ~r  j„i 


cos  dz  =  T-T-r 


N  dy         /Jn 

^  dx"    '    df    '    dz 
dL 
dz 


N  dz      JdV_       d£      djy 
^  dx^  "^  df  "^  dz"    J 


•     (344)' 


CONSTRAINED  MOTION. 


145 


Hence  N  acts  always  in. the  direction  of  the  normal  to  the  devi- 
ating surface. 

Substituting  in  Eqs.  (341)  for  /^r-,  etc.,  their  equals  N  cos  0^^ 


etc.,  we  have,  after  transposing, 


Y-  M~^-^=  -iV^cos  ey\ 


(345) 


Squaring  and  adding,  and  extracting  the  square  root  of  the 
resulting  equation,  we  have 


Representing  the  first  member  by  P  and  dividing  each  of 
Eqs.  (345)  by  (346),  we  have 


X-  M 


y-M^^ 


=  —  cos  dx\ 


—  cos  6 


cos  Sz, 


(347) 


The  first  members  are  the  cosines  of  the  angles  which  the 
resultant  P  makes  with  the  co-ordinate  axes,  and  we  see  there- 
10 


146  MECHANICS  OF  SOLIDS. 

fore  that  P  acts  in  a  direction  opposite  to  the  normal  of  the 
deviating  surface.  We  also  see  that  P  is  the  resultant  of  that 
part  of  the  extraneous  forces  which  generates  no  momentum, 
and  as  its  action-line  makes  an  angle  of  180°  with  the  normal  to 
the  constraining  surface,  it  is  the  measure  of  the  direct  pressure 
on  the  surface;  the  equal  intensity  N  is  the  equivalent  normal 
reaction  of  the  surface.  We  therefore  see  that  the  force  N, 
which  we  have  apparently  introduced,  already  existed  in  the  re- 
action of  the  rigid  surface.  Hence  we  conclude  that  if  a  body 
be  acted  oil  by  any  system  of  extraneous  forces  and  constrained 
to  move  upon  a  rigid  surface,  the  circumstances  of  motion  of 
translation  will  be  precisely  the  same  as  if  it  were  a  free  body 
acted  upon  by  a  system  consisting  of  the  given  forces  and  one 
whose  intensity  and  direction  are  those  of  the  normal  reaction 
of  the  surface.  Equations  (345)  may  therefore  be  employed  in 
problems  of  constraint,  just  as  Eqs.  (Tj^)  are  employed  in  prob- 
lems of  free  motion. 

132.  Transposing  the  second  terms  of  the  first  members  of 
Eqs.  (345)  to  the  second  members,  and  multiplying  the  resulting 
equations  by  dx,  dy^  dz,  respectively,  we  have,  by  addition, 

Xdx  ■\-Ydy-\-Zdz-^NY-^-  cos  Qx-\-~  cos  By  +  ~  cos  e^  j-  ds 

,^dxd^x -^  dydy-{-dzd^z  .     „. 

=^ ^^ ■•  •  •  •  (348) 


which,  since 


dx  Q  ^  dy  /J  ,  ^^ 
-r  cos  Bx-\--j-  cos  By-\--r 
ds  ^    ds  ^  ^   ds 


^  cos  6^  -\-  -^- cos  By  -\-  —cos  6^  =  o .    .    .     (349) 


is  the  cosine  of  the  angle  which  the  normal  reaction  makes  with 
the  tangent  to  the  surface,  reduces  by  integration  to 

J  {Xdx  +  Ydy  +  Zdz)  =  ^^  +  C,  .     .     .     (350) 


CONSTRAINED  MOTION. 


147 


the  equation  of  energy  (126)  of  a  free  body  under  the  action  of 
the  same  forces.  Hence  the  conclusions  derived  from  the 
theorem  of  energy  in  free  motion  of  translation  are  true  in  con- 
strained motion  also  (Arts.  83  and  84);  the  constraint  being 
supposed  to  be  without  friction. 

133.  To  find  the  value  of  N^  eliminate  dt  from  Eqs.  (345)  by 
the  relation 


(351) 


and  we  have 


as 
NCOS  By  =MV'^-  V; 

JVcos  0,  =  MV'^  -  Z. 
as 


(352) 


Squaring  and  adding,  we  have 

yV^'(cos'  e.+  cos'  e,+  cos'  0.)=M'  F'  \  (5)  +  {ff)'+  iff)'  \ 


^x'  -i-  F'4-z^ 


(353) 


Let  p  be  the  radius  of  curvature  of  the  surface  at  any  point, 
and  0  the  angle  which  the  resultant  -ff  makes  with  it;  then  sub- 
stituting in  Eq.  (353)  the  following  values: 

cos'  6jc  -f  cos"  6y  +  cos'  6j  =  1;      .     .     .     (354) 

^»+  K»  +  Z'  =  i?»; (355) 


i?sin0  =  ^^;f  =  ^F'g; 


.     (356) 


148 


MECHANICS  OF  SOLIDS. 


4- 

ds 

d'x 

dx  d's 

ds 

~  ds' 

ds'  ds" 

4 

ds 
ds 

_dy 

~  ds' 

dy  d's^ 
ds'  ds''' 

4 

ds 
ds 

d'z 
~  ds' 

dz  d's 
ds'  ds'' 

the  latter  being  obtained  from  Eq.  (7), — we  have 
,  dx  dy 


dz 
,        I    ds        dz   d's\ 
"^      V  ^^    '^  ds'  ds'/ 


=  H*  sin'  0  + 


M'V 


dx  dy^  ,d_z_  ^ 

,^,,,^X    "^Ts    ,    Y   "^ ds       Z    "^ ds\R 

ds'      {Rds       Rds       Rds  S 

+  R'  cos"  0. 


(357) 


(358) 


-VR' 


(359) 


But  since 


and 


CONSTRAINED  MOTION.  ^49  , 


dx  dy  dz 

X       ds    ^    Y      ds     ^    Z      ds  ,  i  r  \ 


X  dx    ^    Y  dy    ,  Z  dz         .      ,  y^v 


we  have  finally 


^«  ^ .^^'  _  £^Z!^.£2i^  + j?.cos'0+ ^^'sin',^- 2^'sin'0 
="—. -^ r+ie^'cos'^; (362) 


whence 


and 


N  •=. -^  cos  0 (363) 


iV=^cos0 (364) 

I  Therefore  the  normal  reaction  of  the  surface  at  any  point  is 
equal  to  the  difference  between  the  normal  component  of  the 

resultant  of  the  extraneous  forces  and . 

9 
134.  When  the  body  is  in  motion  on  the  concave  side  of  the 
surface  the  value  of  N  is  given  by  Eq.  (363),  and  when  on  the 
convex  side  by  Eq.  (364).  In  the  first  case  we  see  then  that 
when  the  action-line  of  R  lies  outside  of  the  tangent  to  the 
curved  path  of  the  body  the  intensity  of  the  normal  reaction  is 

equal  to  the  sum  of and  R  cos  0,  and  when  it  lies  within 

the  tangent,  to  the  excess  of over  R  cos  0.     When,  in  the 


ISO 


MECHANICS  OF  SOLIDS. 


latter  case,  R  cos  0  becomes  greater  than ,  the  body  will 

leave  £he° concave  surface  and  describe  a  path  of  greater  curva- 
ture, since  iVcan  never  become  negative. 

In  the  second  case  Eq.  (364)  shows  that  the  body  can  only 
remain  on  the  given  surface  as  long  as  R  cos  0  is  positive  and 

numerically  greater  than  — —-\  when  this  condition  is  not  ful- 


Fig.  49. 


filled  the  body  will  leave  the  surface  and  describe  a  path  of  ^ess 

curvature.     Fig.  49  illustrates  the  two  cases. 

135.  Centrifugal  Force. — The  force  whose  intensity  is  meas- 

MV 
ured  by was  formerly  supposed  to  be  exerted   by  the  body 

itself  and  to  act  from  the  centre  of  curvature  outward.  It  was  there- 
fore called  centrifugal  force.  This  name  is  still  in  general  use, 
but  it  is  evidently  a  misnomer  arising  from  erroneous  conclu- 
sions from  well-known  phenomena.  We  have  seen,  Art.  16,  that 
when  the  path  is  a  curve  the  total  acceleration  is  the  resultant 

of  two  rectangular  components,  one,  -3-^,  in  the  direction  of  the 

v^ 
tangent,  and  theother,  — ,  along  the  radius  of  curvature  towards 

P 


CONSTRAINED  MOTION,  15I 

the  centre.  The  intensities  of  the  corresponding  forces  are  there- 
fore  J/ -7-5  and  M — . 

In  the  discussion  above,  we  find  the  latter  force  to  be  that 
part  of  the  normal  reaction  of  rigid  curves  or  surfaces  which  is 
called  into  play  by  the  equal  direct  pressure  due  to  the  change 

in  the  direction  of  motion  of  the  bodv.     Hence is  the  in- 

tensity  of  that  force  which  actually  deflects  the  body  from  its 
rectilinear  path.  It  varies  directly  as  the  square  of  the  velocity 
of  the  body,  and  inversely  as  the  radius  of  curvature  of  its  path; 
it  is  zero  when  p  is  00 ,  that  is,  when  the  path  is  a  right  line,  and 
infinite  when  p  is  zero,  that  is,  no  finite  force  can  abruptly  change 
tlie  direction  of  motion  of  a  body. 

Let  a  body  B,  Fig.  50,  be  whirled  about  a  centre  by  means 
of  a  cord  dB  held  by  the  hand  at  d,  tlie  latter  describing  the 
small  circumference.  The  pull  on  the  cord, 
represented  by  Ba^  has  two  components,  Bb 
and  Bc\  the  former  accelerates  the  motion  of 
B,  while  the  latter  deflects  it  from  the  rectilin- 
ear path  which  it  would  follow  due  to  its 
acquired  velocity  and  the  component  Bb.     If 

the  motion  be  accelerated  until ,  equal  to 

P 
Ba  cos  0,  is  greater  than  can  be  applied  through 
the  medium  of  the  cord,  the  latter  will  break 
and  the  body  will  continue  to  move  in  the 
direction  of  its  motion  at  the  instant  of  rup- 
ture. 

Due  to  the  great  angular  velocities  with 
which  fly-wheels,  grindstones,  etc.,  are  often 
made  to  rotate,  thev  sometimes  break  in  pieces; 

f  '  YiG,   50. 

this  occurs  when  the  cohesion  at  the  sections 
of  rupture  is  less  than  the  pull  required  to  cause  the  parts  of  the, 
body  to  describe  their  circular  paths  with  the  required  velocity. 
"When  the  areas  of  section  of  such  bodies  and  the  values  of  thg. 


152  MECHANICS  OF  SOLIDS. 

cohesion  per  unit  of  area  are  known,  the  safe  values  of  V  can 

readily  be  determined  from  the  expression . 

136.  Referring  to  Art.  44,  we  can  now  see  why  the  rotation 
of  the  earth  on  its  axis  diminishes  the  weight  of  bodies,  and 
makes  the  apparent  less  than  the  actual  weight. 
Since  the  mass  m^  Fig.  51,  is  constrained  to  re- 
main on  the  convex  side  of  its  parallel  of  latitude, 
the  normal  reaction  is 


JV  =  mg  cos  A 

Fig  SI.  =  cos  :i(^  —  Qd'R)m,  .      .       (365) 

I 

in  which  oo  is  the  angular  velocity  of  the  earth,  X  the  latitude  of 
m,  and  p  and  R  the  radii  of  the  parallel  and  of  the  equator 
respectively.  At  the  equator  the  acceleration  due  to  gravity  is 
diminished  by  the  value  of  co^R^  or 

4;r'3962.72  X  5280  ^    .  ,  ,,^ 

which,  as  was  previously  stated,  is  about  ^^  that  due  to  gravity; 
hence  if  the  angular  velocity  of  the  earth  were  seventeen  times 
as  great  as  it  now  is,  bodies  at  the  equator  would  have  no  appar- 
ent weight. 

137.  Problems  in  Constrained  Motion. — To  solve  these  problems 
we  may  either  make  use  of  the  equations  of  constraint  (342), 
together  with  the  equation  of  the  surface,  or  by  means  of  Eqs. 
(345)  treat  them  as  oases  of  free  motion.  By  the  first  method 
we  find  the  partial  differential  coefficients  from  the  equation  of 
the  surface  and  substitute  them,  with  the  component  intensities 
of  the  extraneous  forces,  in  the  equations  of  constraint.  We 
will  have  then  two  equations  involving  three  second  differential 
coefficients,  and  a  third  equation  can  be  obtained  by  differentiat- 
ing the  equation  of  the  surface;  hence  we  can  thus  find  a  single 


CONSTRAINED  MOTION. 


153 


equation  by  their  combination,  involving  only  a  single  second 
differential  coefficient,  its  corresponding  variable  and  constants. 
If  integration  be  possible,  the  solution  of  the  problem  can  then 
be  accomplished.  By  the  second  method  the  substitution  of  the 
normal  reaction  and  the  component  intensities  of  the  extraneous 
forces  in  Eqs.  (345)  gives  three  equations,  each  involving  the  cor- 
responding component  acceleration;  the  steps  are  then  those 
employed  in  cases  of  free  motion. 

138.  On  an  Inclined  Plane. — Let  the  forces  be  friction  and  the 
weight  of  the  body.  Assume  that  the  friction  is  constant  and 
directly  opposed  to  the  motion,  and  let  F  be  its  intensity.  Let 
Mg  be  the  weight  of  the  body,  and  take  the  axis  of  y  in  the  in- 
clined plane,  the  latter  making  the  angle  a  with  the  axis  of  x. 
Let  the  body  start  from  rest  at  the  origin;  the  motion  will  then 
be  in  the  plane  xz,  and  we  have  for  the  equation  of  the  path 


L-=.  z  -\-  X  tan  a  =  o. 


(367) 


Assume  the  second  of  Eqs.  (342),  and    substitute  in  it  the 


following  values: 

dL 

'dz  ~ 

i; 

^ 

dL 

d'x  ~ 

tan  a\ 

d'x  = 

d'z 
tan  Of' 

X=: 

—  i^cos  a\ 

z  = 

^Mg-\-F 

sin  a,  - 

(368) 


This  gives  us  ^  Fig.  52. 

d^z  _  i^  sin  a-\-  F  sin  a  tan'  a  —  Mg  tan'  a 
dt*  ~  M(\-^  tan'  a) 

^   ( 77    '  n^   tan'  a\         .IF  .       \ 

=  M[^^^^oc^Mg^-^  =  sina[-^gsma). 


(369) 


154  MECHANICS  OF  SOLIDS. 

Multiplying  by  2dz  and  integrating,  we  get 


dt 


{F,y  =  2  sin  «r^—  -  g  sin  ajz,  .     .     .     (370) 
or,  solving  with  respect  to  <//, 


4/      •  -^  \  ^^ 

y  2  sin  ol\—  —  g  sin  of  J 

and  by  integration, 

2  =  i  sin  a\j^  -  g  sin  aJ/» (372) 

If  i^  =  o,  we  have 

-s  =  -  igf"  sin"  a (373) 

and 

Vz  =  sin  «  4/2^ (374) 

We  also  readily  get 


^-^  ^  ti^  ^  ^^^  "^  ^^' ^^''^^ 

V=  V{V,Y-^  (V^Y  =  V^z=gfsm  a.     .     .     (376) 


Hence  the  velocity  at  any  point  is  that  due  to  the  height,  and 
varies  directly  as  the  product  of  the  time  and  the  sine  of  the  in- 
clination. 

Comparing  this  value  of  Fwith  that  given  in  the  discussion 
of  motion  due  to  gravity  alone  (Art.  92),  we  see  that  F  is  the 
same  function  of  z,  but  a  different  function  of  /. 


CONSTRAINED  MOTION. 


155 


These  results  can  be  more  readily  obtained  by  considering- 
the  motion  to  be  free,  the  body  being  acted  upon  by  the  result- 
ant of  all  the  forces,  including  the  normal  reaction.  This  re- 
sultant is  Mg  sin  a  —  F,  and  it  acts  in  the  direction  of  the  path. 

139.  Let  the  circumference  OAB  (Fig.  52)  be  in  the  plane 
xz,  and  tangent  to  the  axis  of  x  at  O.  Then  we  have  for  the 
point  A,  Eq.  (372), 


—  2;  =  ^/*  sin*  a  =  OA  sin  a  =  OB  sin"  a  .     ,     (377) 


or 


(378) 


which  is  the  time  of  fall  from  rest  down  the  diameter  of  the 
circle. 

Hence  the  time  required  for  the  body  to  pass  over  any  chord 
of  a  given  circle,  the  plane  of  the  circle  being  vertical  and  the 
body  starting  from  the  upper  extremity  of  the  vertical  diameter, 
is  independent  of  its  length;  that  is,  the  circumference  is  the 
locus  of  simultaneous  arrival  down  all  right  lines  in  a  vertical 
plane,  these  lines  having  a  common  point  at  the  origin  of  motion. 
The  circle  is  called  the  synchronous  curve  of  such  lines. 

From  this  property  of  the  circle  the  right  line  of  quickest 
descent  from  a  given  point  to  a  given  right  line  in  a  vertical 
plane  containing  the  point,  or  from  the 
given  right  line  to  the  given  point,  can 
be  found.  In  the  first  case  let  P,  Fig.  53, 
be  the  point  and  AB  the  line;  draw  the 
horizontal  PA  meeting  the  line  at  A^ 
and  bisect  the  angle  PAB\  the  point  of 
meeting  C  of  the  bisector  and  the  ver- 
tical line  through  P  is  the  centre  of  the 
circle  whose  cord  PO  is  the  required  line.  In  the  second  case 
let  B' A'  be  the  right  line  and  P'  the  point;  draw  the  horizontal 


Fig. 


156 


MECHANICS  OF  SOLIDS. 


F'A'  meeting  B'A[  at  A'\  bisect  the  angle  P'A'B',  and  the  chord 
O'F'  of  the  circle  PO'F'  is  the  required  line. 

140.   On  the  Concave  Side  of  the  Ai'c  of  a  Cycloid  whose  Plane  is 
Vertical. — Take  the  origin  as  in  Fig.   54. 
The  equation  of  the  cycloid  is 


X 


z  i 

X  —  a  versin  -  ^  — |-  (2az  —  z")  •  (379) 
a 


Fig.  54. 


Considering    the    weight    as    the    only 
force,  we  have  X  =  o,  Z  =  —  Mg,  and  Eqs.  (345)  become 


d*x      JV  dz  _ 
IF  ^  His   ~^' 


d'z 


N_dx 
If  Is 


dt""^^       ^^  ^-  ~~  °' 


(380) 


smce 


cos  Qx^ r    and     cos  d^  •=.  -— . 

ds  ds 


Multiplying  by  2dx  and  2dz^  respectively,  and  adding  the  prod- 
ucts, we  obtain 


2dxd^x  +  2dzd'^z 
df' 


+  2gdz  =  o. 


Integrating,  we  have 


dx'  +  dz''       ds'        ,_.  ,    ^ 


(381) 


Supposing  Fto  be  zero  when  z -^^  h,  this  reduces  to 


CONSTRAINED  MOTION.  157 

g  =  r'  =  2g{h  -z)- (38a) 

or,  the  velocity  is  that  due  to  the  vertical  distance  through  which 
the  centre  of  mass  has  fallen. 

The  normal  reaction  is,  Eq.  (363), 


Nz^-j-~RQO-.ct>^—--\-Mg--.    .     .     (383) 


By  differentiating  the  equation  of  the  curve,  we  get 

dx  dz        ds 


(384) 


and  since  the  radius  of  curvature  is 

p  =  \Za(2a  -  ^)]i, (385) 

these  values  give,  for  W, 


^  -  FF-7 m  +  ^S-    i     u       =  ^SV-T Tu  .  (386) 


If  the  body  start  from  rest  at  the  highest  point  of  the  curve, 
then  will  V^  —  2g/i  =  4^^  when  it  reaches  the  lowest  point  ;  then 
z  =  o,  and  we  have 

JV=Mg  +  Mg=2Mg, (387) 

or  double  the  weight  ;  therefore  the  direct  pressure  due  to  the 
velocity  generated  by  the  weight  in  falling  from  the  highest  to  the 
lowest  point  is  equal  to  the  weight  of  the  body. 


158  MECHANICS  OF  SOLIDS. 

Let  the  body  start  from  any  point,  as  P\  then  OP  =  j,  and 
we  have,  Eqs.  (382)  and  (384), 

\       pp       ds       _  fay    M         dz 
{2g)^Jk    W^'^)^~^§l  Jo     (hz  -  zy 

=(-;ri-'"-Ti:-(ir=  •  •  •  •  •  (388) 

that  is,  the  time  of  descent  from  any  point  on  the  curve  to  the 
lowest  point  is  independent  of  its  height  h  and  will  be  the  same 
no  matter  from  what  point  the  body  starts.  The  velocity  with 
which  it  passes  the  lowest  point  will  depend  on  the  vertical 
height  of  fall,  and  is  equal  to  ^ 2gh  :  due  to  this  velocity  the 
body  will  ascend  to  an  equal  height  on  the  other  branch  of  the 

•cycloid  in  a  time  equal  to  that  of  the  descent,  or  Tt\  —  \  ;   after 

which  it  will  return  to  the  point  of  starting,  and  so  on  continu- 
ously.    These  recurring  movements  are  called  oscillations,  and 

since  they  are  performed  in  equal  times,  2;rl— 1  ,   the  cycloid  is 

•called  a  tautochronous  curve. 

The  time  down  any  inclined  right  line  /  is,  Eq   (377), 

^  =  (7^^)*' (389) 

therefore  down  any  radius  of  curvature  of  the  cycloid  it  is 


(390) 


dx 
whence,  substituting  the  value  of  p,  Eq.  (385),  and   of  — ,  Eq. 


CONSTRAINED  MOTION.  1 59 


{384),  and  reducing,  we  get 


=(?)' <-) 


g 

which  is  the  time  of  fall  down  the  maximum  radius  of  curvature, 
or  twice  the  diameter  of  the  generating  circle.  Therefore  the 
times  of  descent  down  all  radii  of  curvature  of  the  cycloid  are 
•equal. 

141.  On  the  Concave  Side  of  a  Circular  Arc  in  a  Vertical  Plane, 
— Let  a  be  the  radius,  and  take  the  origin  at  the  highest  point 
with  the  axis  of  2:  vertical  and  positive  downward.  The  equation 
of  the  circle  is 

X*  =  2az—z*, (392) 

from  which  we  have 

dx         dz       ds 


a  ^  z       X 


(393) 


Let  the  weight  be  the  only  force  acting,  and  denote  the  ve- 
locity at  the  origin  by  V^.     Then  we  have 

MV'-MV:=M{2gz\ (394) 

or 

v'  =  2gz+v: (395) 

For  the  normal  reaction  we  have 

^  =  _..  _  jr^_  =  ^  J.^.  _  Mg^-.  .     (396) 

The  limit  of  N  being  zero,  we  have  for  the  corresponding 
value  of  the  velocity  at  the  origin 

V,  =  V^. (397) 


i6o 


MECHANICS  OF  SOLIDS. 


That  is,  the  body  will  not  follow  the  curve  continuously  in  one 
direction  unless  the  velocity  at  the  highest  point  be  at  least 
equal  to  ^ag.  The  corresponding  value  of  the  velocity  at  the 
lowest  point  is  ^^ag. 

Constrained  Motion  about  a  Fixed  Axis. 


The  Compound  Pendulum. — When  a  rigid  solid  oscillates 
freely  about  a  horizontal  axis  under  the  action  of 
its  weight,  it  is  called  a  compound  pendulum. 
Let  G,  Fig.  55,  be  the  centre  of  gravity,  h  its  dis- 
tance from  the  axis,  and  ^  the  angle  which  h 
makes  with  the  vertical  plane  through  the  axis. 
Then  we  have,  Eq.  (257), 


Fig.  55. 


M. 


Mgh  sin  0 


■  (398) 


or,  taking  ^  so  small  that  it  may  be  substituted  for  its  sine, 


d^^  _  gh^ 


(399) 


Multiplying  by    '2dif)  and    integrating,  supposing    the    body  to 
start  from  rest  when  ip  —  a,  ^q  have 


#'^       S^^      (a'~0')- (4oo> 

and 


dt 


=  i/^ 


-\-  h^      -dtp 
-1         ^/a"  -  f 


(401) 


CONSTRAINED  MOTION  ABOUT  A   FIXED  AXIS.         l6l 


Hence 


d^ 


^         gh      J    |/a»  _  ^»        ^         gh      J      /     _^» 

gh  a  ^       ' 

The  time  of  one  oscillation  is  therefore 

^,  =  ^y  ^^    > (403) 

which  is  the  integral  between  the  limits  ip  =.  a  and  ^  =  —  or. 

The  oscillations  of  a  compound  pendulum  may  therefore  be 
considered  isochronal  when  the  arcs  of  vibration  are  very  small. 
143.  The  Equivalent  Simple  Pendulum. — If  in  Eq.  (403)  we 
make  k^  =  o,  we  shall  have  the  time  of  oscillation  of  a  material 
point  about  a  horizontal  axis  with  which  it  is  connected  by  a 
line  without  weight.  Such  a  pendulum  is  called  a  simple  pendu- 
lum.   Denoting  its  length  by  /,  we  have  for  its  time  of  oscillation 

/'  =  'rV^ (404) 

ii 

For  a  simple  and  a  compound  pendulum  which  are  isochronal 
with  each  other  we  have 

or 

/=^^; (405) 

and  /  is  called  the  equivalent  simple  pendulum  of  the  given  com- 
pound pendulum. 


1 62  MECHANICS  OF  SOLIDS, 

A  point  of  the  compound  pendulum  on  tlie  line  h  at  the  dis- 
tance /  from  the  axis  is  called  the  centre  of  oscillation,  and  a  line 
through  this  centre  and  parallel  to  the  axis  of  suspension  is 
called  the  axis  of  oscillation. 

144.  From  Eq.  (405)  we  have 

{l-h)/i  =  k; (406) 

Hence,  as  regards  their  distances  from  the  centre  of  gravity, 
the  axes  of  suspension  and  oscillation  are  connected  by  the 
same  law  as  the  line  of  impact  and  spontaneous  axis.  Therefore 
the  axes  of  suspension  and  oscillation  are  reciprocal  and  convertible,  and 
the  times  of  oscillation  about  them  are  the  same. 

145.  Adding  2k^  to  both  members  of  Eq.  (406),  we  have, 
after  reduction, 

/=..,  +  (A^* (407) 


The  minimum  value  of  /  is  2/^^,  and  this  occurs  when  h  —  k^ 
r=i  I  —  h.  Therefore,  since  /  is  a  minimum  when  /  is  a  mini- 
mum, the  time  of  oscillation  is  a  minimum  wheri  the  axis  of  suspension 
passes  through  the  centre  of  gyration  7vith  respect  to  an  axis  through  the 
centre  of  gravity  and  parallel  to  the  axis  of  suspension.  It  is  evidently 
a  minimum  minimum  for  that  centre  of  gyration  which  corre- 
sponds to  the  least  central  principal  moment  of  inertia  of  the  body. 

146.  The  Simple  Seconds  Pendulum. — The  simple  seco?ids  pendu- 
lujn  is  a  simple  pendulum  whose  time  of  oscillation  is  one  mean 

cr 

solar  second.     Its  length,  L  —  — 2,  obtained  by  making  /  =  i   in 

Eq.  (404),  varies  directly  with  the  acceleration  due  to  gravity. 

Let  /  be  the  equivalent  simple  pendulum  of  a  given  com- 
pound pendulum,  and  we  have 


CONSTRAINED  MOTION  ABOUT  A  FIXED  AXIS. 


163 


and  substituting  in  the  value  of  Z,  we  get 


^  -  ;r'  -  /- 


{409) 


Hence  to  find  L  it  is  necessary  simply  to  find  the  time  of 
oscillation  of  a  compound  pendulum  and  the  length  of  its  equiva- 
lent simple  pendulum.  To  do  this  Kater 
used  the  pendulum  represented  in  Fig  56. 
Its  centre  of  gravity  is  removed  from  its 
middle  point  by  the  heavy  bob  BB'\  S 
and  O  are  two  knife-edged  prismatic  axes 
of  hardened  steel,  permanently  attached 
to  the  rod  and  having  their  edges  turned 
towards  each  other  ;  w  is  a  small  ring- 
shaped  mass  which  can  be  moved  up  and 
down  so  as  to  change  slightly  the  position 
of  G  and  the  value  of  ^^,  and  is  arranged 
with  clamps  and  a  screw  to  fix  it  in  any  de- 
sired position.  For  any  assumed  position 
of  ni  let  GC  be  the  gyratory  circumference 
which,  from  the  construction  of  the  pen- 
dulum, always  lies  between  S  and  G^  and 
beyond  O  from  G,  as  in  the  figure. 

It  is  evident  that  with  this  arrange- 
ment every  position  of  m  gives  a  different 
compound  pendulum  for  each  axis,  and 
that  there  is  but  one  among  these  whose 
equivalent  simple  pendulum  is  the  length 
h-\-h'  between  the  fixed  axes,  and  for  which 
the  times  of  oscillation  about  6"  and  O  would  be  the  same.  It 
was  this  particular  compound  pendulum  that  Kater  desired  to 
find  experimentally  by  moving  the  sliding  mass  m  until  it  was 
placed  in  the  required  position.  This  he  was  enabled  to  do  by 
the  application  of  the  principles  stated  in  Arts.  144  and  145^ 

By  the  first  principle,  the  times  of  oscillation  about  S  and  O 


Fig.  56. 


164  MECHANICS  OF  SOLIDS. 

are  equal  only  when  these  lines  accurately  coincide  with  the 
axes  of  suspension  and  oscillation.  On  trial  the  times  about 
S  and  O  would  in  general  be  found  to  differ  slightly,  and  by  the 
second  principle  a  displacement  of  m  towards  either  axis  would 
cause  both  times  to  lengthen  or  both  to  shorten,  but  unequally. 
And  since  the  gyratory  circumference  is  nearer  O  than  S,  a 
greater  change  for  the  same  displacement  takes  place  with  re- 
spect to  the  latter  than  to  the  former,  from  the  principle  of  max- 
ima and  minima.  The  pendulum  was  mounted  on  the  axis  S 
and  permitted  to  oscillate  through  small  arcs,  the  number  of 
oscillations,  being  counted;  then  it  was  suspended  from  the  axis 
0  and  oscillated,  the  corresponding  number  in  the  same  time 
being  determined.  Repeated  trials  enabled  Kater  to  find  the 
position  of  m  for  which  the  distance  between  S  and  (9,  or 
// -j- //' z= /,  was  accurately  the  length  of  the  equivalent  simple 
pendulum,  and  its  time  of  oscillation  was  known  from  observa- 
tion. This  method  does  not  require  the  determination  of  the 
exact  position  of  G,  but  has  the  disadvantage  of  exacting  accu- 
rate adjustment  of  the  mass  m,  an  operation  requiring  very  care- 
ful and  repeated  manipulation. 

In  order  to  count  the  number  of  oscillations  the  method  of 
coincidences  was  used.  Thus  the  pendulum  was  mounted  in 
front  of  a  clock  whose  pendulum,  beating  seconds,  could  be  seen 
by  means  of  a  telescope  behind  the  position  of  the  Kater  pendu- 
lum, when  it  passed  the  lowest  point  of  its  arc  of  oscillation.  At 
a  certain  second,  indicated  by  the  clock,  the  two  pendulums 
would  coincide,  and  after  an  exact  number  of  oscillations  of  the 
clock  pendulum  they  would  again  coincide.  The  number  of 
oscillations  of  the  clock  pendulum  in  this  period,  if  the  duration 
of  the  clock  pendulum's  oscillation  was  less  than  that  of  the 
Kater  pendulum,  would  be  two  viore,  and  if  greater  two  less,  than 
the  Kater  pendulum.  The  clock  gives  its  own  indication,  and 
hence  the  other  is  at  once  determined.  In  Kater's  experiment 
the  entire  duration  of  each  trial  lasted  about  thirty-five  minutes, 
corresponding  to  five  coincidences,  or  four  intervals  of  530  sec- 
onds each. 


CONSTRAINED  MOTION  ABOUT  A   FIXED  AXIS.         165 

147.  Another  and  less  tedious  method  is  to  use  a  reversible 
pendulum  of  such  a  form  that  its  centre  of  gravity  may  be  de- 
termined with  considerable  accuracy,  and  whose  axes  are  not 
reciprocal.     Then  we  have,  Eq.  (403), 


k^  -f  /^'  =  Lhf',     \ 

and  eliminating  k^^ 
h^  -h 


L 


=  ht-'   _  ^Y«; 


L~  2  h^h'  '^  2  h-'h'' 


As  the  error  in  this  method  is  due  to  the  approximate  values 
of  h  and  h\  t  and  f  should  be  as  nearly  equal  as  practicable,  and 
h  and  h*  should  differ  as  much  as  practicable,  thus  making  the 
term  which  depends  on  ^  —  h*  very  small, 

148.  The  Value  of  G. — Having  found  L  by  experiment,  the 
acceleration  due  to  gravity  is  found  from 

^=^'^ (412) 

When  g  has  been  determined  for  one  locality  we  may  find  its 
value  for  any  other  place  by  means  of  any  compound  pendulum. 
For  the  two  places  we  have 


-y^-^    and     f  =  n\/-±^..     .     (413) 


1 66  MECHANICS  OF  SOLIDS. 

Hence 

t'U*'v.g^:g (414) 

or 

^  =^5-6" (415) 

If  N  and  JSP  be  the  numbers  of  oscillations  per  hour  at  the 
two  places,  these  become 


N'^:N^v.g^  '.g', (416) 

N 


^'  =  -w^^ (417) 


The  formula  Eq.  (59),  taken  from  Everett's  Units  and  Physi- 
cal Constants,  gives,  for  all  latitudes, 

g  =  32.173  —  0.0821  cos  2/1  —  .000003-^,     .     .     (418) 

pr 

and  from  L  =  —^  we  have  the  corresponding  value  for  the  simple 
seconds  pendulum, 

Z  =  3.2597  —  .0083  cos  2A,  —  .0000003/^.     .     .     (419)' 

Substituting  the  value  of  the  latitude  of  West  Point,  4i°23'3i", 
we  derive 

^  =  32.163/.^.     and     Z=  3.2587//.       .     .     (420) 

for  the  values  of  ^  and  Z  at  West  Point. 

149.  Length  of  the  Equivalent  Simple  Pendulum. — The  length  of 
the  equivalent  simple  pendulum  of  any  compound  pendulum  is, 
Eq.  (409), 

l=Lt\ (42i> 


CONSTRAINED  MOTION  ABOUT  A   FIXED  AXIS.         1 6/ 

in  terms  of  its  time  of  oscillation  and  the  length  of  the  simple 
seconds  pendulum  at  the  place  of  observation. 

150.  British  Standard  of  Length. — In  1824  an  act  of  Parlia- 
ment defined  the  Imperial  Standard  Yard  "  to  be  the  straight  line 
or  distance  between  two  points  in  the  gold  studs  in  the  straight 
brass  rod"  known  as  the  "Standard  Yard,  1760,"  at  62°  F.,  and 
designated  the  ratio  of  its  length  to  that  of  a  simple  pendulum 
vibrating  mean  seconds,  in  vacuo,  at  sea-level  at  the  latitude  of 
London,  to  be  as  Tf^  '•Z9-^Z9Z-  This  standard  was  destroyed  in 
the  burning  of  the  Houses  of  Parliament  in  1834.  Upon  the 
recommendation  of  a  commission  of  scientific  men  appointed  to 
restore  the  standards  of  weights  and  measures,  the  act  of  1855 
defined  the  Imperial  Standard  Yard  oi  Great  Britain  to  be  "the 
straight  line  or  distance  between  the  centres  of  the  two  gold 
plugs  or  pins  in  the  bronze  bar  deposited  in  the  office  of  the 
Exchequer,"  at  62°  F.  Its  restoration  in  case  of  loss  or  destruc- 
tion is  provided  for,  by  reference  to  its  numerous  copies.  The 
present  standard  is  therefore  not  referred  to  tlie  length  of  the 
simple  seconds  pendulum. 

151.  To  determine  the  Mo mefit  of  Inertia  of  a  body  by  the  princi- 
ples of  the  Compound  Pendulum. — It  is  often  necessary  to  find  the 
moment  of  inertia  of  a  body  which  is  not  homogeneous  nor  of  a 
regular  form,  and  to  which  therefore  the  methods  of  Art.  101 
will  not  apply.  Whenever  the  body  can  be  mounted  so  as  to 
oscillate  about  a  horizontal  axis  under  the  action  of  its  own 
weight,  we  may  apply  the  principles  of  the  compound  pendulum 
and  find  its  moment  of  inertia  no  matter  what  its  form  or  sub- 
stance may  be. 

Thus,  multiplying  Eq.  (405)  by  M,  the  mass  of  the  body,  and 
clearing  of  fractions  we  have 

Mlh  =  M{k;  +  h')  =  2mr''] (422) 

that  is,  the  product  of  the  mass  of  the  body,  the  length  of  its  equivalent 
simple  pendulum,  and  tiie  distance  of  the  axis  from  the  centre  of 
gravity,  is  the  moment  of  inertia  with  respect  to  the  axis  about 


1 68  MECHANICS  OF  SOLIDS. 

which  the  body  is  oscillated.  The  first  two  quantities  are  ob- 
tained frcm  the  equations 

W 
M  =  —     and     I  -  Lt\ (423) 

in  which  W  is  the  weight  in  pounds,  ^  the  acceleration  due  to 
gravity  at  the  place  of  observation,  derived  from  Eq.  (418),  Z  the 
length  of  the  simple  seconds  pendulum,  and  t  the  time  of  oscilla- 
tion of  the  body  about  the  axis  in  question.  The  value  of  h  can 
be  found  by  measurement,  provided  the  exact  position  of  the 
centre  of  gravity  is  known.  When  this  is  not  known  h  may  be 
found  as  follows:  Attach  a  dynamometer  to  the 
extremity  farthest  from  the  axis  of  suspension, 
as  in  Fig.  57,  and  take  its  reading  when  the 
I^sbody  has  been  lifted  by  the  dynamometer  to  a 
^^  position  such  that  the  axis  and  centre  of  gravity 

Fig.  57.  are  in  a  horizontal  plane;  this  position  is  reached 

when  the  reading  of  the  dynamometer  is  a  maximum.  Then  if 
R  be  the  reading  in  pounds  and  a  the  horizontal  distance  TS  in 
feet,  we  have,  by  the  equality  of  moments, 

Ra  =  Wh, 
whence 

k=^..  ......  .  (424) 

Substituting  these  values  of  M,  I  and  hy  we  have 

Mlh  =  2n.>'  =  ^.^.Lt'  =  p.   .    .    .    (425) 

To  find  the  moment  of  inertia  with  respect  to  an  axis  through 
the  centre  of  mass,  the  body  must  be  mounted  on  a  parallel  axis, 
with  reference  to  which  its  moment  of  inertia  may  be  found  by 


CONSTRAINED  MOTION  ABOUT  A  FIXED  AXIS.         1 69 

the  above  method.     Its  moment  of  inertia  with  respect  to  the 
axis  through  the  centre  of  mass  is  then 

Mk^  =  Mlh  -  Mh^ (426) 

For  bodies  of  small  mass  it  is  sometimes  more  convenient  to 
attach  them  to  a  pendulum  w^hose  moment  of  inertia  with  respect 
to  its  axis  is  known.  The  moment  of  inertia  of  the  combination 
may  then  be  determined,  and  the  difference  between  this  result 
and  the  moment  of  inertia  of  the  known  pendulum  is  the  mo- 
ment of  inertia  of  the  body  with  respect  to  the  axis  of  the  pen- 
dulum, from  which  the  required  moment  may  be  found. 

152.  The  Conical  Pendulum. — Let  a  simple  pendulum  have  a 
component  vibration  about  two  horizontal  axes  at  right  angles 
to  each  other.  The  path  of  the  material  point  will  in  general  be 
a  curve  of  double  curvature  on  the  surface  of  a  sphere  whose 
radius  is  the  length  of  the  pendulum,  this  length  describing  a 
cone  with  its  vertex  at  the  point  of  suspension.  Such  a  pendu- 
lum is  called  a  conical  pendulum. 

The  equation  of  the  projection  on  a  horizontal  plane  of  the 
curve  of  double  curvature  may  be  determined  as  follows: 

Let  ^  and  0  be  the  arcs  of  oscillation  about  the  two  axes  re- 
spectively; then  sin  ^  and  sin  0  will  be  the  co-ordinates  of  the 
projected  path  of  the  material  point  of  the  pendulum.  If  the 
arcs  be  taken  so  small  that  the  oscillations  may  be  considered 
isochronal,  then  ^  and  0  may  be  taken  as  co-ordinates  of  the 
projected  path,  instead  of  sin  ^  and  sin  0. 

Let  P  and  a  be  the  maximum  values  of  0and  ^  respectively. 
When  0  =  yS  we  will  have  ^  =  o,  and  0  =  o  when  ^  =  or.     Tak- 


which  /  is  measured,  we  have,  Eq.  (402), 


ing  ip  =  a  when  /  =  o,  and  taking  y  —  seconds  as  the  unit  in 

o 


/=cos~^-; (427) 


170  MECHANICS  OF  SOLIDS. 

and  taking  0  =  /?  when  ^  =  o,  or  /  =  \7t,  we  have 


/=cos-^|  +  i;r (428) 


Eliminating  /,  we  get 


cos-i|  =  cos-i-^  +  i;r; (429) 


and  taking  the  cosines  of  both  members, 


(430) 


or 

rfi'i-cP'a'--a^^\ (431) 

the  equation  of  an  ellipse  referred  to  its  centre  and  axes. 

153.  While  the  point   moves   in  azimuth  about   the  vertical 
through  the  point  of  suspension,  the  change  in  azimuth   from 

7t 

the  time  when  tp  ^=  a  until   0  =  /?  is  -.      But   this   deduction   is 

made  under  the  supposition  that  the  vibrations  of  a  simple  pen- 
dulum are  isochronal.  However,  as  the  time  of  vibration  in- 
creases with  the  length  of  the  arc,  the  shorter  component  vibra- 
tion in  the  conical  pendulum  will  be  completed  first,  counting  the 
time  from  any  assumed  epoch.  Therefore  the  change  in  azi- 
muth from  the  time  when  tp  =  a  until  tp  =  —  a  \s  greater   than 

7t,  and  that  from  7p  =  a  until  cp  =  /3  is  greater  than  — .      Hence 

the  axis  of  the  ellipse  has  a  motion  in  azimuth  in  the  same  di- 
rection as  that  of  the  pendulum. 

A  closer  approximation  shows  that  the  change  in  azimuth 


EQ  UILIBRIUM.  1 7 1 


7t  (  ■?  \  ^ 

from  ^  =  a  until  0  =  /?  is  -( i  +  f^/^j  instead  of  -.     The  el- 
lipse  therefore  makes  one   complete  revolution  in  azimuth  in 

8       .  .  .    J.      • 

— -  times  Its  periodic  time. 
Za6 


Equilibrium. 

154.  When  a  body  is  in  a  state  of  equilibrium,  Art.  68,  the 
acceleration  factors  in  the  general  equation  of  energy  become 
separately  equal  to  zero,  and  there  can  then  be  no  change  of  po- 
tential into  kinetic  energy,  or  the  reverse.  Therefore  during 
equilibrium  the  general  equation  of  energy  reduces  to 

2Idp  =  o.     ,     : (S) 

In  general  a  free  body  is  never  in  a  state  of  equilibrium,  since 
it  is  subjected  to  the  action  of  forces  the  resultant  of  which  is  not 
in  general  zero.  Hence  a  body  in  equilibrium  must  be  under 
constraint.  But  since  any  case  of  constrained  motion  may  be 
discussed  as  one  of  free  motion  by  introducing  the  normal  reac- 
tion of  the  constraining  curve,  the  general  equation  of  energy 
may  thus  be  made  to  apply  to  all  cases  of  equilibrium. 

155.  There  are  three  cases  in  which  the  acceleration  may  be- 
come zero,  viz.: 

(i)  When  the  resultant  is  zero,  the  body  being  at  rest  or  hav- 
ing uniform  motion. 

(2)  When  the  resultant  of  the  system  of  forces  reverses  its 
direction  as  it  passes  through  zero,  thus  changing  the  sign  of  the 
resultant  acceleration. 

(3)  When  the  resultant  becomes  zero  but  does  not  pass 
through  it;  in  this  case  there  is  no  change  of  sign  of  the  result- 
ant acceleration. 

First.  When  the  body  is  at  rest  the  forces  are  called  siresseSy 
and    they  produce  changes  of   form  and   volume,  these  effects 


1/2  MECHANICS  OF  SOLIDS. 

usually  being  known  as  strains;  their  investigation  properly  be- 
longs to  Applied  Mechanics. 

When  the  accelerations  become  zero  because  of  the  uniform 
motion  of  the  body,  Eq.  (S)  simply  asserts  that  the  quantity  of 
work  done  positively  by  some  of  the  forces  is  equal  to  that  done 
negatively  by  the  others,  the  whole  quantity  of  energy  added  to 
the  system  being  zero.  If  a  body  be  supposed  at  rest  at  any 
point  of  that  portion  of  its  path  over  which  it  has  uniform  mo- 
tion, it  will  evidently  remain  there  if  subjected  only  to  the  sys- 
tem of  forces  which  caused  it  to  follow  that  path.  In  sucli  a 
•case  the  body  is  said  to  be  in  neutral  or  indifferent  equilibrium. 

156.  To  investigate  the  second  csiSQ  let  us  resume  Eq.  (132). 
The  forces  of  gravitation,  electricity,  etc.,  or  what  are  known 
generally  -as  forces  of  nature,  are  taken  to  be  constant  or  to  vary 
as  some  function  of  the  distance;  and  therefore  Eq.  (132)  is  ap- 
plicable to  a  body  subjected  to  their  action,  the  normal  reaction 
of  the  curve  on  which  the  body  moves  being  considered  as  one 
of  the  extraneous  forces. 

Assuming  consecutive  values  of  the  kinetic  energy  of  the 
body,  we  have,  after  developing  the  difference  of  the  correspond- 
ing, states  of  the  function  by  Taylor's  theorem, 

WV:  -  WV:  =  n^,  +  dx^,  y^  4-  dy^,  z^  +  ^5,)-  F{x^,  y^,  z^} 
=  Xdx,+  Ydy^  +  Zdz^±{Adx^^-\-  Bdy^-^  Cdz^)-\-  etc.  (432) 

If  \MV^  be  a  maximum  or  a  minimum,  we  shall  have  as  a 
condition 

Xdx,  -f  Ydy^  +  Z^^,  =  o; (433) 

that  is,  the  body  will  be  in  equilibrium  when  it  reaches  a  position 
where  it  has  a  maximum  or  minimum  kinetic  energy. 
Let  this  condition  be  fulfilled,  and  we  have 

\MV^  -  \MV^  =  ±  {Adx^  +  Bdy^'  +  Cdz^)  -f  etc.    (434) 


EQUILIBRIUM.  1 75 


If  \MV^  be  a  maximum  the  second  member  of  this  equation 
will  have  the  negative  sign,  and  whatever  be  the  value  of  V^,  it 
must  be  greater  than  V ^ ;  and  if  the  body  be  slightly  displaced 
from  its  position  of  equilibrium  and  then  move  from  rest  under 
the  action  of  the  given  system  of  forces,  the  direction  of  the  re- 
sultant must  be  such  as  to  bring  it  back  again.  In  this  case  the 
body  would  oscillate  to  and  fro  through  its  position  of  equilib- 
rium, and  could  never  depart  far  from  it;  the  equilibrium  of  a 
body  is  therefore  said  to  be  stable  when  it  occupies  a  position, 
corresponding  to  a  maximum  value  of  its  kinetic  energy. 

If  \MV^  be  a  minimum  the  second  member  of  Eq.  (434)  will 
be  positive,  and  hence  V^  must  be  greater  than  F,  ;  and  if  the 
body  move  from  rest  and  from  a  point  very  near  that  corre- 
sponding to  the  minimum  value  of  F,  the  resultant  of  the  system 
will  act  in  such  a  direction  as  to  move  it  away  from  its  position 
of  equilibrium,  to  which  it  would  never  return.  The  equilibrium 
is  therefore  said  to  be  unstable  when  it  occupies  a  position  corre- 
sponding to  a  minimum  value  of  its  kinetic  energy. 

157.  In  the  third  case,  since  a  function  is  not  necessarily  a 
maximum  or  a  minimum  when  its  differential  coefficient  is  equal 
to  zero,  it  is  evident  that  cases  may  arise  in  which  a  body  will  be 
in  equilibrium  when  its  kinetic  energy  is  neither  a  maximum  nor 
a  minimum.  For  example,  a  body  is  in  equilibrium  as  it  passes 
a  point  of  inflection  at  which  the  resultant  is  normal  to  the  path. 
In  this  case  the  equilibrium  is  stable  in  one  direction  and  un- 
stable in  the  other. 

158.  When  2ifree  body  passes  a  point  in  its  path  correspond- 
ing to  a  maximum  or  a  minimum  value  of  its  kinetic  energy,  this 
point  would  be  a  position  of  stable  or  unstable  equilibrium,  if  we 
suppose  the  body  to  be  moving  on  a  rigid  curve  coincident  with 
its  path. 

Let  us  take  the  general  case  of  a  body  acted  upon  by  its  own 
weight,  and  subjected  to  any  condition  of  constraint  whatever. 
Since  this  condition  can  have  no  influence  on  the  velocity,  the 
change  of  kinetic  energy  between  any  two  points  of   the  path 


174  MECHANICS  OF  SOLIDS. 

will  be  that  due  to  the  weight  acting  over  the  vertical  distance 
through  which  the  body  moves;  hence  we  may  write 

WV;  -  mV^  =  Mg{z,  -  ^.),  ....     (435) 

z  being  taken  vertical  and  positive  downward.  From  this  we  see 
that  if  the  body  be  in  stable  equilibrium  its  centre  of  mass  must 
occupy  a  point  in  its  path  which  is  lower  than  the  consecutive 
points  on  either  side;  and  similarly  if  it  be  in  unstable  equilib- 
rium it  must  be  at  a  point  which  is  higher  than  the  consecutive 
points. 

Thus,  a  pendulum  is  in  stable  equilibrium  when  its  centre  of 
gravity  occupies  the  lowest  point  of  its  possible  path,  and  in  un- 
stable equilibrium  when  it  is  at  the  highest  point  it  can  reach 
under  the  given  conditions  of  constraint.  A  homogeneous  el- 
lipsoid of  three  unequal  axes,  resting  on  a  horizontal  plane,  has 
two  positions  of  unstable  and  one  of  stable  equilibrium;  an 
oblate  spheroid  has  one  of  stable  and  many  of  unstable  equilib- 
rium; a  prolate  spheroid  has  one  of  unstable  and  many  of  stable 
equilibrium,  and  a  sphere  is  an  example  of  indifferent  equilib- 
rium. If  the  centre  of  gravity  of  the  sphere  be  not  coincident 
with  the  centre  of  figure,  it  will  be  in  stable  equilibrium  when 
the  centre  of  gravity  is  at  the  lowest  point  it  can  reach,  and  un- 
stable when  at  the  highest. 

Examples,  i.  A  particle  on  the  concave  surface  of  a  sphere 
is  acted  on  by  its  weight  and  by  a  repulsion  from  the  lowest 
point  of  the  surface,  the  latter  varying  inversely  as  the  square  of 
the  distance.     Find  the  position  of  rest. 

Take  the  origin  at  the  lowest  point  and  the  axis  of  z  vertical 
and  positive  upward;  let  r  be  the  distance  of  the  point  of  rest 
from  the  origin,  a  the  radius  of  the  surface,  and  ^  the  intensity 
of  the  repulsive  force  at  the  distance  unity;  then  we  have  the 
equation  of  the  surface, 

^^  +y  +  -2;'  —  2az  =  o; 


EQUILIBRIUM. 


175 


the  intensity  of  the  repulsion  at  the  distance  r, 


and  Eqs.  (T^,), 


2az 


x  =  J^^ 


2az '  r 


N' 


o; 


N^-  =0; 


2az  r  a 


(436) 


from  which  we  have,  substituting  in  the   last  of  Eqs.  (436)  the 
value  of  N  obtained  from  one  of  the  others  and  reducing, 


^  =  ^     and 


w 


2       aw 


(437) 


that  is,  the  particle  will  remain  at  rest  at  any  point  in  the  cir- 
cumference of-fi  horizontal  circle  whose  plane  is  at  the  distance 
given  by  Eq.  (437)  above  the  lowest  point  of  the  surface. 

If  another  repellent  force  whose  intensity  at  a  unit's  distance 
is  /i'  be  supposed  to  act  on  the  particle,  the  value  for  /-'  would 


be  r''  = 


7a  ' 


hence  the  ratio  of  the  intensities  of  these  forces  at 


or  their  intensities  are 


the  distance  unitv  is  given  by  —  = 

directly  as  the  cubes  of  the  distances  at  which  a  heavy  particle 
would  remain  at  rest  on  the  surface  of  a  sphere  due  to  their 
action.  The  quadrant  electroscope,  consisting  of  a  light  pith 
ball  joined  to  a  point  by  a  thread,  measures  the  relative  inten- 
sities of  strong  electrical  charges  by  the  principles  of  this  prob- 
lem. 


176 


MECHANICS  OF  SOLIDS. 


2.  To  find  the  position  of  rest  of  a  heavy  particle  m  on  a. 
given  rigid  curve  AB^  Fig.  58,  when  acted  on 
by  its  weight  w  and  a  constant  attraction  t 
toward  the  origin.  Let  O  be  the  origin,  Ox  and 
Oz  the  co-ordinate  axes,  the  latter  being  vertical 
and  positive  downwards,  and  /;/  the  supposed 
position   of    rest    of    the    particle;    let    OA  =  a, 

Then  we  have 


Fig.  58. 


Om  =  r,  and  mOn  =  6. 


X=  -/sin  6  -\-N 


ds 


Z=ze/  —  /cos  d  —  N 


dx 
Ts 


o: 


(438) 


from  which  we  get 

{w  —  t  cos  6)dz  —  /  sin  6dx  =  wdz  —  / =  o.     (439) 


But 

xdx  4"  zdz  =  rdr, 

and  Eq.  (439)  reduces  to 

wdz  —  tdr  =  o,  . 


(440) 


(441) 


which  is  the  condition  of  equilibrium.  In  order  that  there  may 
be  a  position  of  rest  on  the  curve  this  condition  must  be  satisfied 
by  the  co-ordinates  of  one  of  its  points. 

Let  the  curve  be  a  hyperbola,  and  we  have 


^V  -  «V  =  a'd^; (442) 

^''  =  ^-^+^«=:.V-^'; (443) 

rdr  =  e^zdz; (444)' 


EQ  U I  LIBRIUM.  I  *J*J 

and  Eq.  (441)  becomes 

rwdz  —  te^zdz  =  o (44S) 

Solving  Eq.  (443)  with  respect  to  z  and  substituting  in  the  result- 
ing equation  the  value  of  r^  obtained  from  Eq.  (445),  we  have 


The  equilibrium  in  this  case  evidently  requires  w  '>et. 

If  it   be  required    to   find  the  equation  of  the  curve  on  all 
points  of  which  m  will  be  at  rest,  we  have,  from  Eq.  (441), 

wz  —  tr  =  3.  constant, (447) 

which  may  be  written 

wz  —  tr  —  {w  —  t)a (448) 

By  substituting  for  z  its  value  /-  cos  6,  this  equation  becomes 


(■-?) 


^  = ' (449) 

I cos  6 


the  polar  equation  of  an  ellipse,  parabola,  or  hyperbola,  accord- 
ing as  w  is  less  than,  equal  to,  or  greater  than  /,  the  pole  being 
at  the  focus. 
12 


178  MECHANICS  OF  SOLIDS. 


^  The    Potential.  • 

159,  The  general  theory  of  attraction  embraces  the  consider- 
ation of  those  forces  by  which  matter  attracts  or  repels  other 
matter,  and  whose  intensities  are  functions  of  the  masses,  and  of 
the  distances  which  separate  them.  The  general  term  attraction 
is  used  for  both  repellent  and  attractive  forces;  the  former  being 
affected  by  the  positive  and  the  latter  by  the  negative  sign,  to 
distinguish  them. 

In  the  following  discussion  the  theory  of  attraction  is  limited 
to  those  forces  whose  intensities  are  expressed  by  the  law,  Eq.  (2), 

^       mm' 

Electrical  and  magnetic  forces  are  sometimes  attractive  and 
sometimes  repellent;  and  since  their  intensities  vary  according 
to  the  law  of  the  inverse  square  of  the  distance,  Eq.  (2)  may 
also  be  applied  to  them  by  considering  m  and  m'  to  be  quantities 
of  free  electricity  or  magnetism  instead  of  masses;  the  unit  quantity 
being  that  which  will  attract  an  opposite  or  repel  a  similar  unit 
quantity  with  a  unit  intensity,  at  a  unit's  distance  apart.  In  the 
following  discussion  the  term  mass  will  then  for  convenience  be 
taken  to  apply  to  the  quantities  m  and  m\  As  in  gravitation, 
these  mutual  attractions  or  repulsions  are  equal  in  intensity, 
opposite  in  direction,  and  exert  their  own  influence  whether 
other  forces  act  on  the  masses  or  not. 

160.  Component  Attractions. — Let  m  be  the  attracted  mass,  m' 
one  of  the  attracting  masses,  r  the  distance  between  them,  and 
X,  J,  z  the  co-ordinates  of  m'  referred  to  m.     We  have  then 

^«  =  ^«+y  +  ^«; (450) 

dr  _x       dr  _y        dr  _  z  .       . 

d^~P     ~^~7'     ~d^~~r'    '     '     '     '     ^^^'^ 


■&)=-.. 


^^ (452) 


THE  POTENTIAL. 


179 


The  component  attraction  of  m*  for  m  in  the  direction  of  the 
co-ordinate  axis  x  will  then  be 


mm'    X 
X  =  — 7[-U  —  =  —  mm  -. 

r        r  dr    ax 


(453) 


and  similarly,  for  the  other  axes, 


.,/(^).  ,-    ..,'(^) 


K=  —  mm*ii-^\    Z 
dy 


mm  }i 


dz 


(454) 


The  sums  of  the  component  attractions  of  all  the  molecules 
m\  m'\  m"'y  etc.,  for  tn  are  therefore 


-y  = 


(  4)     -^W     ^ 

_,„^\^„'._  +  ^"__  +  etc.y 


<f5- 


—  w/l- 


dx 


d:2 


m' 


y=  —  mfji- 


dy 


Z=  —  w/(- 


(455) 


which,  when  we  place 


(456) 


i8o 


MECHANICS  OF  SOLIDS, 


become 


Y  = 
Z  = 


—  mp. 


mjji 


mfx 


dx'' 
dj/ 

iV 
dz' 


(457) 


That  is,  the  sum  of  the  component  attractions  of  the  masses  m\  m'\ 
etc.,  on  m,  in  any  direction,  is  equal  to  the  corresponding  partial  differ- 
ential coefficient  of  the  function  V  7?tultiplied  by  the  product  mfx. 

l6l.  The  Potential. — The  function  Miscalled  th^  Potential  oi 
the  mass  2m'  with  reference  to  thQ position  of  m,  and  is  defined 
to  be  the  sum  of  the  quotients  obtained  by  dividing  the  mass 
or  corresponding  quantity  of  each  element  m'  by  its  distance 
from  m. 

To  explain  what  is  meant  by  the  potential,  let  dn  be  the 
change  in  the  potential  energy  of  a  unit  mass  attracted  by  m\ 
under  the  assumed  law  of  attraction,  when  the  unit  mass  has 
changed  its  distance  by  dr\  then  we  have 


dn  = 


m'}x 


(458) 


whence 


->/-  ^ 


m'  fx 


+  0, 


(459) 


or,  between  the  limits  r  and  co , 


,      /''■       dr 


m*  II 


.  (460) 


THE  POTENTIAL.  l8l 


If  the  unit  mass  be  subjected  to   the   attraction   of   all  the 
masses  in\  m'\  etc.,  then  we  have 


^n  ^  ^-2-- =  ixV, (461) 

r 

which  becomes,  when  the  absolute  intensity  }a  is  taken  to  be 
unity^ 

27r=:2—=  F. (462) 


That  is,  the  potential  is  equal  to  the  change  in  the  potential  energy  of 

a  unit  mass  when  the  latter  is  moved  to  infinite  distance  from  any  dis- 

m* 
tance  r  against  the  decreasing  attraction  —  -2— j,  or  when  the  unit  mas^ 

r 

is  brought  from  infinite  distance  to  any  distance  r  in  opposition  to  the  in- 

creasing  repulsion^  2-^  \  the  absolute  intensity  being  taken  as  unity  in 
r 

both  cases. 

For  any  definite  system  of  masses  and  forces  governed  by 
the  assumed  law  the  potential  at  a  point  has  a  definite  value  and 
can  be  expressed  in  units  of  work.  When  the  potential  is  known 
the  component  attraction  in  any  direction  can  be  readily  ob- 
tained by  means  of  Eqs.  (457). 

162.  Equi'potential  Surfaces. — Let  R  be  the  intensity  of  the 
resultant  attraction  of  the  system  of  masses  for  m  at  the  dis- 
tance r\  then,  after  multiplying  Eqs.  (457)  by  dx^  dy,  dz,  respec- 
tively, and  adding,  we  have 

fdV  dV  dV    \ 

Xdx  +  Ydy  -\-  Zdz  =  —  mpiy—dx  -\-  -j-dy  +  —j-^A^  (4^3) 

or 

Rdr  =  —  jnfidV^ (464) 


1 82  MECHANICS  OF  SOLIDS. 


whence 

P  r^   —  ^m  IJ. 

dr 


dV 

^  =  -  "^i^iz (465) 


Let  s  be  the  path  of  m  as  it  changes  its  position  in  any  direc- 
tion; d  the  angle  which  s  makes  with  r\  and  /the  component  of 
R  in  the  direction  of  s.     Then  we  have 

I  ■=^  R  cos  cy  =  —  m)j.——  cos  U  =.  —  in}x—-. .     .     (466) 


Hence  the  component  attraction  in  any  direction  varies  directly  with 
the  first  differential  coefficient  of  the  potential  regarded  as  a  function 
of  the  path  in  that  direction. 

An  equi-potential  surface  is  one  for  which  the  potential  is  con- 
stant for  each  of  its  points.  If  the  path  j  be  a  line  of  such  a  sur- 
face, we  have 

Ids  Rdr  . 

dV  — = =  o, (467) 

m^  mjx  \-r  1/ 

and  hence 

F=  -  —  fids  =  -—  fRdr  =C.    .     .     (468) 

A  surface  which  fulfils  for  each  of  its  points  this  condition 
is  an  equi-potential  surface  for  the  system  of  attractions.  As 
any  value  may  be  attributed  to  C  between  its  greatest  and  least 
values,  there  will  be  an  indefinitely  great  number  of  equi-poten- 
tial surfaces,  corresponding  to  any  given  system  of  attractions^ 
each  of  which  will  be  a  closed  surface. 

From  Eq.  (466)  it  is  evident  that /becomes  zero  when  ^  is 
90°,  and  is  the  resultant  attraction,  or  R^  when  Q  is  zero;  hence 


THE  POTENTIAL.  1 83 


the  equi-potential  surface  cuts  at  right  angles  the  direction  of  R 
at  every  point  of  its  surface.  These  action  lines  of  R  are  called 
lines  of  force,  and  any  collection  of  them  passing  through  an 
elementary  portion  of  the  surface  is  called  a  tube  of  force. 

If  the  surface  be  supposed  perfectly  smooth,  vi  would  remain 
at  rest  on  every  point  of  it  if  subjected  only  to  the  system  of  at- 
tractions, and  the  surface  would  resist  pressure  only  in  the  di- 
rection of  the  normal.  For  this  reason  it  is  called  a  level  surf  ace 
or  a  surface  of  equilibrium.  No  two  surfaces  belonging  to  the 
same  system  can  intersect  or  have  a  common  point,  for  Eq.  (468) 
cannot  be  satisfied  for  the  same  values  of  x,  y,  z,  and  give  C 
dissimilar  values.  Of  any  two  surfaces,  the  interior  one  corre- 
sponds to  the  greater  resultant  attraction  and  the  less  value  of 
the  potential  when  the  attraction  is  negative,  and  to  the  greater 
repulsion  and  greater  value  of  the  potential  when  the  attraction 
IS  positive. 

163.  The  determination  of  the  values  of  the  potential  and  the 
attractions  for  any  given  system  of  masses  or  quantities  2m^ 
depends  on  the  solution  of  the  equations 


V=  2--,     R=-  nifx—     and     /  =  -  w/^-^.       (469) 


When  the   quantities  w'  are  elements  of  a  quantity   whose 
density  and  boundary  vary  by  continuity,  we  may  write 


m'  =  dM'  =  ddv; (47°) 

in  which  dv  represents  the  elementary  volume  of  M\     In   such 
cases  we  have,  when  rectangular  co-ordinates  are  employed, 

^^jSJ^^jSdxdydz ^^^^^ 


1 84 


MECHANICS  OF  SOLIDS. 


If  it  be  desirable  to  use  polar  co-ordinates,  we  have  for  the 
z  value  of  the   elementary  volume, 

Fig.  59, 

=  r'dr  sin  OdcpdO;      ]   ^"^^^^ 

for  the  edge  of  the  infinitesimal 
cube  in  the  direction  of  r  is  dr, 
the  horizontal  edge  perpendicular 
to  r  is  r  sin  6dcp,  and  that  per- 
pendicular to  the  plane  of  these  two  is  rdS;  and  hence,  in  polar 
co-ordinates, 


=    /  drdr  sin  6d(pd6 (473) 


164.  Examples. — In  the  following  examples  the  mass  m  will 
be  taken  to  be  the  unit  mass,  and  the  absolute  intensity  fx  to  be  unity 
also.  By  multiplying  the  results  obtained  by  ?nfJL  we  readily  get 
the  attractions  for  any  mass  and  any  intensity. 

I.    The  Potential  and  Attractions  of  straight  Rod  at  an  External 
Point. — Let  go  be  the  area  of  the  rod's  cross-section,  y  the  dis- 
tance of  the  external  point  m,  Fig.  60,  from 
the  rod,  and  r  its  distance  from  any  element 
of  the  rod;    and  let  the  axis  of  the  rod   be  a 

taken    as    the   axis   x,   and    the   axis  y   pass   ° 
through    the    external    point.     The    element  ^"^-  ^° 

volume  of  the  rod  is  then  oodx^  and  the  distance  r  =  Vy^  -\-  x'^; 
then  we  have 


V=Sgd  T- 


=  =(y&?iog^.  ^  ,,        i  .   (474) 


4//  +  X 


THE  POTENTIAL.  1 85 


which,  taken  between  the  limits  x**  and  —  x\  corresponding  to 
the  extremities  of  the  rod,  becomes 


=  ,Ja,log-^-^^== (475) 


The  component  attraction  in  the  direction  of  the  rod  is 

To  find  that  perpendicular  to  the  rod  we  have,  from  Eq.  (474), 


dy  dy 


Therefore  between  the  limits  ^"  and  —  x'  we  have 


i86 


MECHANICS  OF  SOLIDS. 


When  the  point  m  is  on  the  perpendicular  bisecting  the  rod, 
we  have 

X  =  o\     Y= .——===  = sine';.     .      (479) 


in  which  6  is  the  angle  included  between  the  bisecting  perpen- 
dicular and  the  line  drawn  from  m  to  the  extremity  of  the  rod. 
2.  The  Potential  and  Attractions  of  a  Circular  Arc  at  its  Centre. 
— Let  d  be  tlie  angle  subtended  by  any 
portion  of  the  arc  estimated  from  its  mid- 
dle point,  Fig.  61.  The  element  volume  is. 
then  oordd,  and  we  have 


V=dGD  r  dd  =  26006', 


(480) 


which  is  independent  of  the  radius  of  the 
arc.  The  resultant  attraction  is  evidently  in  the  direction  CO^ 
or  along  the  radius  drawn  to  its  middle  point.     Its  value  is 


doo 


L 


^'  rdO 


2600 
r 


(481) 


From  this  we  see  that  the  attraction  of  the  arc  at  the  centre  is. 
the  same  as  the  attraction  of  the  straight  rod  AOA' .  Since  also 
the  masses  of  the  elements//'  and  PP'  have  the  ratio 


CP        

Cp  :CPstca  =  Cj>  :  CP^  =  C/  :  CP\   .     .     (482). 


their  attractions  on  m  at  Care  equal;  whence,  any  portion  of  the 
right  line  tangent  to  the  arc  at  O,  as  PP\  attracts  m  at  C  with 
the  same  intensity  as  the  corresponding  arc//'. 


THE  POTENTIAL.  \%J 


3.  The  Potential  and  Attraction  of  a  uniform  Circular  Ring  at 
a  Point  on  the  Perpendicular  to  its  Plane  through  its  Centre. — Let  a 
be  the  radius  of  the  ring,  and  c  the  dis- 
tance of  the  point  (9,  Fig.  62.     Then 

•^'^'  27taSG0  ,    „    , 

V=  2—  =     ^  .     (483) 

r         ^a'  +  c'  ^     ^ 

When   c  =  o,    this   becomes  a  maxi- 
mum, or  27rSco,  which,  being  independent  Fig.  62. 
of  the  radius,  shows  that  Fis  a  constant  for  all  concentric  rings 
at  the  centre. 

As  the  sum  of  the  component  attractions  of  the  elements  of 
the  ring  in  the  plane  of  the  ring  is  zero,  the  resultant  attraction 
is  in  the  direction  of  the  perpendicular  to  the  plane  of  the  ring.. 
Its  value  is  evidently 

R=-2^co%QOC  =  -,-2m'  =  j^^-j^^    .     .     (484) 

4.  The  Potential  and  Attraction  of  a  Circular  Plate  at  a  Point 
on  the  Perpendicular  to  its  Plane  through  its  Centre. — Let  /  be  the 
thickness  of  the  plate,  Fig.  62,  and  suppose  the  plate  made  up  of 
separate  rings  whose  width  is  da.  Then  we  have  00  =  tda,  and 
the  element  volume  is  27ttada\  whence 

V=2n6tr-^^^  =  2nSt(\fT-:^-c\,.     (485) 

t/o      ya  -\-  c 

which  for  the  centre  of  the  plate  becomes 

F=  2ndta (486) 

The  resultant  attraction  is 

2  7ttada  y^^r^  A      P^      ada 


R 


n^  2ntada  /^       aa 

=  01     5 —  cos  QOC  =  27tdtc  /     —_ — 

e/o  r  J^      V^-f  c^ 

=  27ttc6[ ,       ^         X=27t6t{l ^         -V       .       (487) 


1 88  MECHANICS  OF  SOLIDS. 


But  since 


^a''  +  c 


=  =  cos  QOC, 


it  follows  that  the  attraction  of  all  circular  plates  of  the  same 
thickness  and  density,  for  a  given  molecule  on  a  perpendicular 
to  its  plane  through  its  centre,  is  the  same  for  equal  angles,  sub- 
tended by  the  plate  at  the  molecule.  Therefore,  if  a  molecule 
be  at  the  vertex  of  a  right  cone  with  a  circular  base,  the  attrac- 
tions of  the  normal  sections  of  equal  thickness  of  the  cone  for 
the  molecule  are  equal.  If  the  radius  of  the  plate  be  infinite  the 
attraction  becomes  27cdi,  which  is  independent  of  the  position  of 
m  with  respect  to  the  plate. 

5.  The  Potential  and  Attractions  of  a  Spherical  Shell  at  any 
Point. — Taking  the  centre  of  the  shell  as  the  origin  of  a  system 
of  polar  co-ordinates,  let  a  be  the  radius  and  p  the  distance  of 
the  point  from  the  centre;  then  the  volume  element  is,  Eq.  (472), 

aV  sin  ddddcf), 
and 

r=z{a''  -  2ap  cos  6  +  py-,      ....     (488) 

whence 

cv    o   /»*''   n""  sin  Odd  del) 

Integrating  first  with  respect  to  0,  we  have 

-    ,   />Tr  sin  Odd  .       . 

V=27t6ta-^       — ;      .     .     (490) 

e/o    (a^  —  2ap  cos  6  -\-  pry 

and  then  with  respect  to  6, 


THE  POTENTIAL.  1 89 


There  are  two  cases  to  consider:  (i)  when  m  is  within  the 
surface,  or  p  <  «;  and  (2)  when  m  is  without  the  surface,  or  p  >  «. 
In  the  first  case  we  have 

r=^— -^[«  +  p- (^-p)]  =  4^(^/«;    .     .     (492) 


and  in  the  second. 


[«  +  p-(p-«)]  =———  =  --;    (494) 


9    "- P  9 

tdta'  _  A 
dp  p^  ~  p^ 


■7;:  =  -^  =  ^-7^  =  --^'^  • (495) 


M  being  the  mass  of  the  shell.  Hence  the  potential  of  the 
interior  space  is  constant  and  the  resultant  attraction  zero,  while 
all  external  space  is  made  up  of  concentric  spherical  equi-poten- 
tial  surfaces,  and  the  attraction  at  any  point  is  the  same  in 
intensity  and  direction  as  though  the  whole  quantity  M  were 
concentrated  at  the  centre  of  the  shell. 

6.  The  Potential  and  Attraction  of  a  Thick  Homogeneous  Spheri- 
cal Shell  at  any  Point. — Let  the  radii  of  the  exterior  and  interior 
surfaces  of  the  shell  be  a'  and  a"  respectively.  The  potential 
will,  in  general,  consist  of  two  parts,  one  corresponding  to  the 
shell  within  the  spherical  surface  containing  the  point  and  the 
other  to  the  shell  without  it. 

For  the  first  part  we  have  (Ex.  5) 

dV='-I^da: (496) 

and  for  the  second, 

dV  =  47rdada; (497) 


IQO  MECHANICS  OF  SOLIDS. 

Jience 

A.7td     f*^  />*' 

V  = /    a^da  4-  attS  I    ada 

P    Ja"  J? 

If  the  point  be  wholly  without  the  shell,  then,  M  being  the 
mass  of  the  shell,  we  have 


V 


=  4^^  r'a^^a  =  i^^(«-  -  an  =  ^;        (499) 

P   J  a"  3P  ^  P  ^    ^' 


^  =  ^; (500) 

and  if  wholly  within, 

V  =  27td{a"  -  a"')', (501) 

^  =  o (502) 

If  the  attracting  quantity  be  a  homogeneous  sphere  a"  =  o, 
>and  at  an  interior  point 

V=,„Sa"-'-^^; (S03) 

Ii  =  ±nSp  =  ^p- (S04) 


o 


a 


and  at  an  exterior  point 


^=f (507) 

p  ^> 


THE  POTENTIAL. 


191 


For  p  =.  a'  vje,  have,  from  both  sets, 

r=4^;    .     .     (508)  i?=^;    .     .    .     (S09) 

hence  both  Fand  R  are  continuous  functions. 

From  this  we  see  that,  considering  the  earth  as  homogeneous, 
we  may  take  its  potential  and  attraction  at  any  external  point 
as  though  the  whole  mass  of  the  earth  were  concentrated  at  its 
centre;  while  the  attraction  at  an  interior  point  is  directly  pro- 
portional to  its  distance  from  the  centre. 

165.  The  Theorem  of  Laplace. — Let  S  be  any  closed  surface, 
and  M  any  attracting  quantity  wholly  external  to  S\  let  V  be 
the  potential  of  J/,  and  p  the  normal  to  the  surface  reckoned 
outward.     Then  it  is  to  be  proved  that 


ndV 


dS  -  o. 


(510) 


From  any  molecule  w',  Fig.  63,  of  M  draw  a  right  line  pierc- 
ing the  surface  S.     It  will  pierce  the  surface  in  an  even  number 


Fig.  63. 


of  points.     Let/'  and/"  be  a  pair  of  these  points,  6'  and  ^"  the 
angles  made  by  the  normals  at  /'  and  /"  with  the  intersecting 


192  MECHANICS  OF  SOLIDS. 

line,  and  r'  and  r"  the  distances  of/'  and/"  from  m\     Then  we 
have 


With  m'  as  a  vertex,  conceive  an  indefinitely  small  cone  whose 
solid  angle  is  cd  to  be  described  about  w'/'.  The  areas  inter- 
cepted by  the  cone  on  S  at  the  points/'  and/"  are 

dS'  =  -^,     and     ^y  =  -^-;     .     .     .     (512) 
cos  d'  cos  6"  ^'^     ' 

and  hence 


^J^,dS'  =  m'm;    ^-dS"  =  -  m'm;     ....     (513) 

g:.5'+^/5'-o; (5x4) 

and  this  result  is  true  for  every  pair  of  points. 

Now  suppose  the  cone  whose  vertex  is  w'  envelops  the  whole 

surface  S;   then  its  solid  angle  is  made  up  of  an  indefinitely 

great  number  of  elementary  cones  whose  solid  angles  are  c»,  to 

,each  of  which   the  above  reasoning  will  apply.     Therefore  we 

have 


/ 


^■5=»- fe"> 


166.  Poisson's  Extension  of  Laplace's  Theorem. — If  any  portion 
of  M,  as  M\  be  contained  within  the  closed  surface  6",  then  it  is 
to  be  proved  that 


/ 


'l^dS=-A7tM' (516) 


THE  POTENTIAL.  1 93 


Let  m'  be  the  mass  of  one  of  the  molecules  of  M\  and  let  a 
right  line  be  drawn  from  it,  piercing  S.  It  will  pierce  S  in  an 
odd  number  of  points,  which  may  be  arranged  in  pairs  as  above, 
with  one  point  remaining. 

For  the  pairs  Ecj.  (515)  will  be  true,  and  for  the  odd  point 
we  have 

;^'  =  -"^- •  •  (517) 

Integrating  for  the  whole  space  about  w',  we  have 

—dS=—Pt'l     doD  =  —  ^7tm'\  .     .     .     (518) 
and  for  all  the  molecules  of  J/', 


/ 


'^-f^dS=-4nM' (S19) 


Hence  we  conclude,  Eqs.  (515)  and  (519),  that  the  sum  of  the 
attractions  of  a  mass  J/,  estimated  along  the  normals  at  all 
points  of  a  closed  surface,  is  zero  when  the  attracting  matter  M 
is  wholly  external  to  Sy  and  is  —  ^7tM'  when  the  closed  surface 
S  contains  any  portion,  J/',  of  M. 

167.  This  theorem  is  also  expressed  by  the  equations 

d'V   ,   d'V   .   d'V  ^  /       X 

-d^^W^~dF  =  ''     ^^      =-A7t6.,     .     (520) 

To  show  this,  let  x,  y,  z  be  the  co-ordinates  of  the  attracting 
molecule,  of  density  (J,  without  or  within  the  surface,  which  may 
be  taken  as  the  surface  of  a  rectangular  parallelopipedon  dxdydz. 
Then 


/ 


dp 


13 


194  MECHANICS  OF  SOLIDS. 


for  the  face  dydz^  is 


and  for  the  opposite  face  is  • 

Sd'^v.    ,dv), 

-and  for  this  pair  of  faces  is 

-j^dxdydz', 

and  similarly  for  the  other  faces  we  have 

d'^V 
-—^dxdydz 

and 

d'^V 

a  dxdydz. 

Hence  the  integral 

Placing  the  second  member  equal  to  o  and  —  ^nM'  in  succes- 
sion, we  have,  when  the  attracting  matter  is  external  to  a  closed 
surface, 

d^V   ,   d'V       d'V  .       . 


MOTION  OF  A    SYSTEM  OF  BODIES. 


195 


and  when  it  contains  it,  wholly  or  in  part, 


(522) 


These  theorems  find  their  most  frequent  application  in  elec- 
tricity and  magnetism. 


Motion  of  a  System  of  Bodies. 

168.  The  conclusions  of  Arts.  78  and  82  with  respect  to  the 
motion  of  a  single  body  under  the  action  of  extraneous  forces 
are  similarly  true  of  a  group  or  system  of  bodies  when  the  mo- 
tion of  its  centre  of  mass  determines  the  translation  of  the 
system  in  space,  and  the  rotation  is  estimated  about  that  centre. 
To  show  this,  let  x^,y^,z^be  the  co-ordinates  of  the  centre  of 
mass  of  the  system  referred  to  any  fixed  origin;  x,y,  z,  the  co- 
ordinates of  the  centre  of  mass  of  each  body  referred  to  the 
fixed  origin,  and  x\y\  z\  when  referred  to  the  centre  of  mass 
of  the  system;  and  let  M  be  the  type-symbol  of  the  masses  of 
the  bodies.  Applying  Eqs.  (T^,)  to  each  mass  and  summing  the 
results,  we  have 


^X  =  2M' 


df 


:EY=:2Mg, 


2z  =  :2M 


dt"" 


(523) 


and  since 


x  =  x,-\-x\  y=y^  +/,  zz=^z,-\-z\  .     (524) 

d^x  =  d'x^-\-dW,   dy  =  dy^-{-dy,   d'z  =  d\+d*z',  (525) 


196 


MECHANICS  OF  SOLIDS. 


these  become,  by  applying  the  principle  of  the  centre  of  mass, 


2V 


2Z  = 


df         ' 


d' 


dt' 


'-2M. 


(526) 


Similarly  applying  Eqs.  (Tm')  when   impulsions   alone   act,  we 
have 


at 

dz 

-^^M  =  2MF.; 


(527) 


hence  the  conclusions  of  Art.  78  follow  from  Eqs.  (526)  and  (527) 
with  respect  to  the  centre  of  mass  of  the  system. 

169.  If  each  of  Eqs.  (526)  be  multiplied  in  succession  by  the 
two  co-ordinates  which  it  does  not  contain,  and  the  difference  of 
the  products  be  taken,  then  the  summation  of  these  differences 
for  all  the  bodies  of  the  system  gives 


:s(y,,-jr,,)  =  2Jf(..5'/-,,5<) 


2(Xz,  -  Zx,) 
2(Zy,  -  Yz,) 


d^x^  d"^  z 


■  (528) 


from  which  the  motion  of  the  centre  of  mass  about  the  fixed 
origin  may  be  found. 


MOTION  OF  A    SYSTEM  OF  BODIES. 


197 


In  the  same  way  we  have,  from  Eqs.  (Tm), 


(529) 


Substituting  in  these  Eqs.  (529)  the  values  of  the  co-ordinates 
and  accelerations  from  Eqs.  (524)  and  (525)  and  reducing  by 
the  principle  of  the  centre  of  mass,  we  obtain 


2{  Yx'  -  xy)  =  2M  (^'^  -y^); 


(530) 


Since  these  equations  are  independent  of  the  co-ordinates  of 
the  centre  of  mass  of  the  system,  we  conclude  that  the  motion 
of  rotation  of  the  centres  of  mass  of  the  constituent  bodies  about 
the  centre  of  mass  of  the  system  is  the  same  as  if  this  point  were 
at  rest,  and  their  motion  is  therefore  entirely  independent  of 
the  motion  of  translation  of  the  latter  point,  a  conclusion  pre- 
cisely similar  to  that  of  Art.  82. 

170.  Conservation  of  t/ie  Motion  of  the  Centre  of  Mass, — If  we 
suppose  that  a  material  system  has  been  put  in  motion  and 
then  subjected  only  to  the  mutual  attractions  of  its  own  bodies, 
we  shall  have 


2x  =  o,   :2Y^o,   :ez  = 


(531) 


198  MECHANICS  OF  SOLIDS. 

then  there  can  be  no  accelerations  of  the  centre  of  mass  of  the 
system,  and  Eqs.  (526)  become 


d^'x,  d^y,  d'^z, 

df      '    dt'      '    dt' 


o; .   .   .   .   {532) 


from  which  we  have 


sc,-  at-Y  a\      y^=z  bt-Y  b\       z^  =  a  +  c';.     .     (534) 

- (535) 


x^-a'  _  y^  -  b'  _  z^ 


a  b  c 

That  is,  if  a  system  of  masses  be  subjected  only  to  its  mutual  attrac- 
tions^ its  centre  of  7nass  will  either  be  at  rest  or  more  uniformly  in  a 
right  line.  This  is  called  the  principle  of  the  conservation  of  the 
motion  of  the  centre  of  mass. 

171.  If  the  masses  of  the  solar  system  be  subjected  only  to 
their  mutual  attractions  of  gravitation,  the  conditions  of  Eqs. 
(531)  are  satisfied  for  this  system,  and  therefore  its  centre  of 
mass  must  have  uniform  and  rectilinear  motion,  or  be  at  rest. 
Since  the  mass  of  the  sun  is  very  much  greater  than  the  sum 
of  all  the  other  masses  of  the  constituents  of  the  system,  the 
error  of  assuming  the  centre  of  mass  of  the  solar  system  to 
be  coincident  with  that  of  the  sun  is  slight.  Calculations  found- 
ed on  the  observations  of  astronomers  show  that  this  latter  point 
is  moving  through  space  with  a  velocity  of  very  nearly  five  miles 
per  second,  but  sufficient  data  is  not  yet  available  to  determine 
whether  its  path  is  a  right  line  or  an  arc  of  small  curvature;  the 
latter  being  the  more  probable,  owing  to  the  extraneous  forces 
of  attraction  of  other  systems. 

By  the  same  principle,  the   motion   of  the  centre  of  mass  of 


MOTION  OF  A    SYSTEM  OF  BODIES. 


199 


the  earth  is  uninfluenced  by  earthquakes  or  volcanic  explosions 
occurring  upon  it,  and  that  of  the  centre  of  mass  of  a  projectile 
is  not  affected  by  its  explosion,  since  the  impulsive  forces  in  each 
of  these  cases  are  mutually  counterbalanced. 

172.  Conservation  of  Moments,  Invariable  Axis,  and  Invariable 
Plane. — If  the  forces  acting  on  the  system  be  the  mutual  attrac- 
tions of  its  masses,  we  have  the  conditions 


:2{Yx*  -  Xy')  =  o: 
2{Xz'  -  Zx')  =  o; 
2{Z/   -  Vz')  =  o; 


(536) 


which  reduce,  Eqs.  (530),  to 


2M\ 
2M\ 


and  which,  by  integration,  become 


>  , 


(S37) 


at 

:^M  -         -  c   , 

at 


(S38) 


That  is,  when  the  forces  acting  on  the  system  are  the  mutual 
attractions  of  its  masses,  the  algebraic  sums  of  the  moments  of  the 
momenta  of  the  masses  of  the  systetn  with  respect  to  any  set  of  rectangu- 
lar co-ordinate  axes  at  the  centre  of  mass  of  the  system  are  constant; 
this  is  called  the  principle  of  the  conservation  of  moments. 


200  MECHANICS  OF  SOLIDS. 

This  principle  may  also  be  stated  as  follows:  If  the  bodies  of 
the  system  be  supposed  at  rest  in  any  one  of  its  configurations, 
a  definite  system  of  impulsions  would  give  each  body  its  actual 
velocity.  Eqs.  (538)  show  that  the  sum  of  the  component  mo- 
ments of  these  impulsions  with  respect  to  each  co-ordinate  axis 
is  constant.  The  resultant  moment  of  the  system  is  also  con- 
stant, and  is  given  by 


C  =  VC  +  C  +  C' (539) 

As  the  co-ordinate  planes  may  be  assumed  at  pleasure,  it  is 
evident  that  the  constants  C,  C",  C"  will  in  general  change 
with  each  set  of  co-ordinate  axes.  The  resultant  axis  of  the  sys- 
tem on  which  C  would  be  measured  is  normal  to  that  plane  with 
reference  to  which  the  sum  of  the  products  of  the  projected 
areas  by  the  masses  is  the  maximum  constant,  and  is  the  com- 
mon intersection  of  those  planes  on  which  these  sums  are  zero. 
This  axis,  and  the  normal  plane  through  the  centre  of  mass  of 
the  system,  are  called  the  invariable  axis  and  invariable  plane  of 
the  system  of  masses;  the  equation  of  the  latter, 

Cz'  +  cy  +  c"v  =  o, .  .  .  .   .   (540) 

is  found  by  multiplying  each  of  Eqs.  (538)  by  the  co-ordinate 
which  it  does  not  contain  and  adding  the  results  together. 

173.  Conservation  of  Areas. — Eqs.  (538)  express  another  prin- 
ciple, which  is  known  as  the  conservation  of  areas.  Let  radii- 
vectores  r  be  drawn  from  the  centre  of  mass  of  each  body  to  that 
of  the  system,  supposed  at  rest ;  then  changing  x'dy'—y'dx'  into 
its  equivalent  expression  in  polar  co-ordinates  with  the  pole  at 
the  centre  of  mass  of  the  system,  we  have 

x'  =■  r  cos  6,         dx'  =  dr  cos  0  —  r  sin  ddd\  \  ^       ^ 

y  =  r  sin  e,         d/  =  dr  sin  e  +  r  cos  edd;  j      *     ^^"^^^ 

x'dy'  -  fdx'  =  r\iB (542) 


MOTION  OF  A   SYSTEM  OF  BODIES.  201 

But  r^dd  is  twice  the  projection  on  the  plane  3(^y'  of  the  sectoral 
area  described  by  the  radius  vector  of  M*  in  the  time  dt\  and  the 
corresponding  factors  in  the  other  equations  are  similarly  the 
projections  of  double  the  differential  areas  on  the  planes  x*z*  and 
y'z*  respectively.  Let  the  type  symbols  of  twice  these  projec- 
tions on  the  planes  x'y\  x'z\  y'z'  be  denoted  respectively  by 
dAz,  dAy^  and  dA^ ;  Eqs.  (538)  then  become 


at 

2M^^  =  C"; 
at 

if  A 
at 


(543) 


Inregrating  between  the  limits  corresponding  to  the  interval  /, 
we  have 

'2MA,  =6-7;^ 

:2MAy  =  C't',    I (544) 

:2MA^  =  C"V. 


That  is,  if  a  system  of  masses  be  subjected  only  to  its  mutual  attrac- 
tions^ the  sum  of  the  products  of  each  mass  by  the  projection  of  its  secto- 
ral area  about  the  centre  of  the  system  on  any  plane  varies  directly 
with  the  time.  This  statement  of  the  principle  is  called  the  con- 
servation of  areas. 

If  the  resultant  of  a  system  of  extraneous  forces  act  through 
the  centre  of  the  system,  Eqs.  (536)  will  be  satisfied  and  the  con- 
clusions of  Arts.  170,  172  will  apply  to  this  case  also. 

174.  Relative  Acceleration. — If  one  of  two  bodies  be  supposed 
fixed  and  all  the  motion  be  attributed  to  the  other,  the  accelera- 
tion which  the  latter  would  have  under  this  supposition  is  called 
its  relative  acceleration.  To  find  the  relative  acceleration  of  one  body 
of  the  system  with  reference  to  the  centre  of  mass  of  any  other,  let 


202  MECHANICS  OF  SOLIDS. 

Mbe  the  mass  to  which  the  motion  is  referred,  and  call  this  body 
the  central;  M\  the  mass  of  the  moving  body,  called  l\\^  primary \ 
M'\  the  type-symbol  of  the  masses  of  the  remaining  bodies  of 
the  system,  called  Xh^  perturbating  bodies.  Let  the  symbol  (AfM^) 
represent  the  intensity  of  the  reciprocal  attraction  of  the  masses 
M  and  J/'  along  the  line  joining  their  centres,  and  the  same 
symbol  with  a  subscript  letter,  as  (MM')^,  the  component  inten- 
sity in  the  direction  of  the  corresponding  axis;  and  similarly  for 
the  other  masses  and  directions.  Let  x,  y,  z  be  the  co-ordinates 
of  J/'  referred  to  M.  The  component  relative  acceleration  of 
J/'  with  respect  to  Mis  the  sum  of  their  actual  component 
accelerations  due  to  their  mutual  attraction,  plus  the  difference 
between  the  components  of  their  actual  accelerations  due  to  the 
attractions  of  the  perturbating  bodies.    The  actual  accelerations 

of  M  and  J/'  due   to  their  mutual  attraction  are  ^ — r^r-—  and 

M 

(MM')        ,     ,         ,  ,    ,  ,.     .        ^    .. 

-^^ — T77   »  and   those  due  to  one  of  the  perturbatmg  bodies  are 

(MM'')       ,  (M'M")     ^,  ,     .  ,        . 

^^ — TF—    and  -^^ —    ,    \    The  component  relative  accelerations  are 

therefore 


d'xr(MM%     (MM^-l     r2{MM")^     2(M'M")^ 
df~V      M      '^     M'     J  +  L        M  M' 

dy_  r{MM%    {MM')y-\    r:E(MM")y  _  :2{M'M")y 

df'V      M      '^      M'    J  +  L        M  M' 

d'z_\-(MM'),  ^  {MM')r\  ,  y^{MM'%     2{M'M") 
dt 


+ 

V{MM'),      iMM%-\     r2(MM'%  _  :2{M'M"),-] 
\_      M      ^      M'     J  +  L        M  M'        J 


^(545) 


175.  The  path  of  the  centre  of  mass  of  the  primary  with 
respect  to  the  central  is  called  the  relative  orbit  of  the  primary  ; 
its  relative  path  influenced  by  the  action  of  the  perturbating 
bodies  is  called  the  disturbed  or  actual  orbit,  and  if  the  action  of 
these  latter  bodies  be  neglected  the  resulting  relative  path  is 


CENTRAL  FORCES. 


203 


called  the  undisturbed  orbit.     The  differential  equations  in  this 
case  become 


df        + 

d\ 

-df=^ 


(MM')       {MM')-lx    ^ 


M 


+ 


M 


P]?, 


\MM')       {MM'y 

.     J/       '^~   M'  " 

)    (^^j/'r 


V{MM' 


(546) 


in  which  the  upper  sign  corresponds  to  attraction  and  the  lowti 
to  repulsion,  and  r  is  the  radius  vector  of  M' .  When  the  law  ot 
the  reciprocal  attraction  or  repulsion  is  known  the  value  of  its 
intensity  may  be  substituted  for  its  symbol  {MM'),  and  the 
resulting  equations  being  integrated  twice,  there  will  result  the 
component  relative  velocities  and  co-ordinates  of  the  centre  of 
mass  of  the  primary  referred  to  the  centre  of  mass  of  the  central 
body. 


Central  Forces. 

176.  A  central  force  is  one  whose  action-line  is  directed  to  or 
from  a  fixed  point  called  the  ce7itre  of  force,  and  whose  intensity 
is  a  function  of  the  distance  of  the  body  acted  on  from  that 
point.  The  force  is  attractive  or  repulsive  according  as  its 
action-line  is  directed  toward  or  from  the  centre. 

177.  Laws  of  Central  Forces. — (i)  Let  the  two  masses  M  and 
M'  be  subjected  to  the  action  of  their  mutual  attraction  or  re- 
pulsion. Then  the  motion  of  one,  relative  to  the  centre  of  mass 
of  the  other,  may  be  considered  as  resulting  from  the  action  of 
a  central  force  whose  centre  is  the  centre  of  mass  of  that  body 
which  is  considered  as  fixed;  Eqs.  (546)  are  then  applicable. 
Multiply  the  first  by  j  and  the  second  by  x,  and  take  the  differ- 
ence of  the  products;  then  multiply  the  first  by  z  and  the  third 
by  Xy  taking  the  difference  of  the  products;  and  lastly,  multiply 


204 


MECHANICS  OF  SOLIDS. 


the  second  by  z  and  the  third  by_>',  taking  the  difference  of  the 
products,  and  we  shall  obtain 


//> 


d-'x 


''ae 

^-d¥ 

— 

o; 

d'x 

d'z 

''dt^- 

= 

o; 

d'z 

^dt^ 

^dt' 

= 

o. 

*       •       • 


(S47) 


Integrating,  we  have 


dy         dx       ,, 

dt       -^dt 

dx        dz        ,,, 

'dt-''-di  =  ''  ' 

^dt  dt 


(548) 


Multiplying  these  equations  by  z,  y  and  x^  respectively,  we  have, 
by  addition, 

h'z-\-h''y-\-h'"x  =  o, (549) 

the  equation  of  an  invariable  plane.  Hence  the  orbit  of  a  body 
acted  on  by  a  central  force  is  contained  in  a  fixed  plane  through  the 
centre  of  force. 

(2)  Take  xy  to  be  the  plane  of  the  orbit;   then  Eqs.  (548) 
reduce  to  the  single  equation 


dy         dx       .  ,       V 


or,  in  polar  co-ordinates. 


,dQ 
dt 


=  >^; 


(551) 


CENTRAL  FORCES.  20$ 


in  which  r  is  the  radius  vector  of  M\  6  is  the  variable  angle 
made  by  r  with  a  fixed  line  of  reference,  and  h  is  double  the 
sectoral  area  described  by  the  radius  vector  in  its  own  plane  in 
the  unit  time. 

Integrating  Eq.  (551)  between  the  limits  corresponding  to 
values  of  the  radius  vector  r'  and  r"  and  the  angles  6'  and  6'\ 
we  have 

r'^'r^dd^hif  ^f')  =  ht',    ....     (552) 

dr"9" 

or,  the  sectoral  area  described  by  the  radius  vector  in  the  plane  of  the 
orbit  varies  directly  with  the  time. 

Reciprocally,  when  this  law  is  fulfilled  we  have,  by  differen- 
tiating Eq.  (550)  and  multiplying  by  M\ 

M'^,x-M'^y=Yx-Xy  =  o;.    '.    .    (533) 


and  the  orbit  is  described  under  the  action  of  a  central  force. 

(3)  From  Eq.  (551)  we  have 

du        h  /       X 

or,  the  angular  velocity  of  the  body  in  its  orbital  motion  about  the  centre 
of  force  varies  inversely  as  the  square  of  the  radius  vector, 

(4)  Since  the  velocity  of  the  body  in  its  orbit  is 

ds       ds  dd  ,       V 


we  have,  from  Eq.  (554), 


h    ds  I        r\ 

''  =  PTff (SS6) 


206  MECHANICS  OF  SOLIDS. 

Let/  be  the  length  of  the  perpendicular  from  the  centre  "of 
force  to  the  tangent  to  the  orbit  at  the  body's  place;  then 


(557) 


.      n          rdd 
^  =  r  sm  0  =  r-^, 

and  hence 

h 

""      P' 

(558) 


or,  the  velocity  of  the  body  varies  inversely  as  the  perpendicular  distance 
fro7ti  the  centre  of  force  to  the  tangent  to  the  orbit  at  the  place  of  the 
body. 

(5)  Let  A  be  the  relative  acceleration  at  any  point  of  the 
orbit,  and  p  the  corresponding  radius  of  curvature;  then  the 
component  relative  acceleration  in  the  direction  of  p  is 

^^  =  -^'; (559) 


from  which  we  have 


4-M •  (560) 


But  2p—  is  the  length  of  the  chord  of  curvature  drawn  through 

the  centre  of  force  to  the  place  of  the  body.  Comparing  Eq. 
(560)  with  V"^  =  2ghf  we  see  that  the  actual  velocity  of  the  body  at 
any  point  of  its  orbit  is  that  due  to  a  height  equal  to  one  fourth  of  the 
chord  of  curvature  drawn  through  the  centre  of  force  ;  the  body  starting 
from  rest  and  the  intensity  of  the  central  force  remaining  constant  over 
this  distance.  If  the  orbit  be  circular,  R  its  radius,  and  its  centre 
coincide  with  the  centre  of  force,  the  velocity  becomes  constant 
and  is 

V'^AR (561) 


CENTRAL  FORCES.  20/ 


The  acceleration  in  the  direction  of  the  tangent  to  the  orbit 
is 

_  =  _  ^_    .     ....     .     .     .     (s6.) 

Multiplying  by  M'  and  ds  and  integrating  between  limits,  we 
have 

W{V.'-V,')  =  -M'£'Adr,     .     .     .     (563) 


the  equation  of  energy.  Hence  the  orbital  velocity  is  inde- 
pendent of  the  path  described  and  varies  with  the  distance  of 
the  body  from  the  centre  of  force.  In  any  closed  orbit,  therefore, 
when  the  body  returns  successively  to  the  same  position  in  its 
orbit,  the  velocity  will  always  be  the  same  as  before. 

These  are  the  general  laws  of  central  forces,  and  are  seen  to 
be  independent  of  the  character  and  law  of  variation  of  the  cen- 
tral force. 

178.  The  Differential  Equation  of  the  Orbit. — Assuming  the  co- 
ordinate plane  xy  as  the  plane  of  the  orbit  and  employing  polar 
co-ordinates,  we  have,  from  the  law  of  areas,  Eq.  (551), 


which  becomes,  when  r  is  replaced  by  — , 

^^f'"' (S64) 

Differentiating 

n         COS  6  /    ,    X 

x-rcose=  ; (565) 


208  MECHANICS  OF  SOLIDS. 

and  dividing  by  dt^  we  obtain 

dt  u"  'dt 


=  —  hiu  sin  6  +  cos  6-\     ....     (566) 


and  hence 


=  -/^v(«cos^  +  cos^^).    .     .    .     (567) 

Placing  this  value  equal  to  that  of  the  relative  acceleration 

d^x 

—--  in  the  first  of  Eqs.  (546),  we  have,  after  dividing  by  cos  d 

X 

and  remembering  that  —  =  cos  ^, 

{MM')    ,   {MM')         .       -^  u,  J     ^   ^M  ,  ^ox 

the  differential  polar  equation  of  the  orbit  of  a  body  under  the 
action  of  a  central  force. 

179.  To  solve  the  direct  and  inverse  problems  in  the  case  of 
a  central  force  we  proceed  as  follows: 

(i)  To  find  the  equation  of  the  orbity  substitute  for  ^  in  Eq. 
(568)  its  value  in  terms  of  «,  and  integrate  twice;  the  resulting 
equation  expressing  the  relation  between  u  and  B  is  the  equation 
of  the  orbit.  The  two  arbitrary  constants  which  appear  in  the 
integration  are  determined  by  the  initial  conditions,  viz.,  the 
initial  values  of  the  radius  vector  and  velocity,  and  the  initial 
direction  of  motion. 

(2)  To  find  the  law  of  the  force  necessary  to  cause  a  body  to 


CENTRAL  FORCES.  209 


describe  a  given  orbit,  differentiate   the  polar  equation  of  the 

orbit  twice  and  substitute  the  resulting  value  of  — ^  in  Eq.  (568); 

then  eliminate  6^,  and  the  result  will  give  ^  in  terms  of  r. 

180.  Particular  Cases  of  the  Direct  Problem, — (i)  To  find  the 
orbit  due  to  an  attractive  central  force  whose  intensity  varies  directly 
with  the  distance  of  the  body  from  the  centre.  Let  the  centre  of 
force  be  the  origin;  }x\  the  measure  of  the  intensity  of  the  central 
attraction  for  a  unit  mass  at  a  unit's  distance.  From  the  law  of 
the  force  and  the  differential  equation  of  the  orbit  we  have 


or 


^=/'''-=^'-M"+^) (569) 


d^u  a' 


Multiplying  by  idu  and  integrating,  we  get 


5^  +  "  =-^'  +  ^- (57') 


Let  jR  be  the  initial  value  of  the  radius  vector,  and  take  the 
initial  direction  of  motion   perpendicular  to  ^.     Then  at  the 

time  /  =  o  we  have  ^  =  o  and  «  =  -  ;    hence  ^  =  ipi  +  ^ , 
and 

^  +  «   -^»+-;^-^i.  .     .     .     .     (572) 

Let  Fbe  the  initial  value  of  the  velocity,  and  from  Eq.  (558) 
we  have  h  :=  RV,     Substituting  this  value  of  h  in   Eq.  (572), 
14 


2IO  MECHANICS  OF  SOLIDS. 

multiplying  through  by  u^^  and  changing  the  form  of  the  result- 
ing equation,  we  have 

Solving  with  reference  to  dQ^  and  multiplying  by  2,  we  have 

ludu 


2de- 


^    \       2V'R'      I  \  2V'R'      I 

2  V'R' 

,2udu 


.     (574) 


,   '^  ^      V  y  -  m'R'         I 

Integrating  between  limits  corresponding  to  /  =  o  and  /,  re- 
membering that  the  initial  radius  vector  coincides  with  7?,  we 
have 

„         .         2V'R''u''  -{V^  ^'R-")         .  .       , 

2^  =  sm-^ ___Jy_^ ^-sin-^i;    (575) 

whence  we  have 

sin  (90° +  26/)  =  cos  2^  = --^—^Jl L,    (576) 

Clearing  of  fractions  and  changing  to  rectangular  co-ordi- 
nates, we  have,  recalling  that 


cos""  6  =  -^-^- — 3     and     sin"  6  =  -—--    , 


V 


2  V'S'  -{V'  +  lx'IP){x''  +/)  =  (V'-  m'H^Xx'  -/);  (577) 


CENTRAL  FORCES,  211 

whence  we  have 

;^^  +  ^  =  i, (578) 

the  equation  of  an  ellipse  referred  to  its  centre  and  axes;  hence 
the  orbit  of  a  body  acted  on  by  a  central  attraction  varying  directly  with 
the  dista?ice,  is  an  ellipse  whose  centre  is  at  the  centre  of  force. 

To  find  the  velocity  at  any  point  of  the  orbit.  Let  v  be  the  gen- 
eral value  of  the  velocity  of  the  body  in  its  orbit;  then  from 
Eq.  (558)  we  have 

h^ 
^'=J.' (579) 

and  from  Eq.  (556)  we  get 

I   _    ds"    _r'de''-^dr'  _  du^ 

Hence 

But  from  Eq.  (572)  we  have 


and  therefore 

V' =  V  +  ix'{R- -  r'), (583) 

which  gives  v  when  r  is  known. 

To  find  the  Periodic  Time.     The  semi-axes  of  the  orbit  are, 
y 
E<^-  (578),  J^  and   -—=:     The  periodic  time,  or  the  time  required 


212  MECHANICS  OF  SOLIDS. 

for  the  body  to  complete  its  orbit,  is,  by  the  law  of  areas,  therefore 

V 

"[^-^^  _2nRV__jn_ 


Hence  the  periodic  time  is  independent  of  the  dimensions  of  the 
orbit. 

Examples  of  orbits  under  this  law  are  found  in  molecular  vi- 
brations; in  the  small  vibrations  of  elastic  bodies,  such  as  tun- 
ing-forks, stretched  strings,  etc.;  and  in  the  oscillations  of  a 
pendulum  through  small  arcs. 

181.  (2)  If  the  central  force  be  repellent,  the  discussion  above 
may  be  made  applicable  by  changing  the  sign  of  /i'  in  Eqs.  (568) 
and  (578);  the  equation  of  the  orbit  then  becomes 


^  =  .; (585) 


hence  the  orbit  is  an  hyperbola  whose  centre  is  at  the  centre  of  force, 

182.  (3)   To  find  the  orbit  when  the  central  force  is  attractive  and 
varies  inversely  as  the  square  of  the  distance.     Assume  the  same  no- 
tation as  in  the  previous  problem,  and  let  the  direction  of  the 
initial  velocity  make  any  angle  with  the  prime  radius  vector  H. 
Then  we  have 


^  =  ^'  =  ,,V  =  /5v(J  +  «),      .    .    .    (586) 


or 

dd"    '  '^~  h^ 


%^  =  ^ (587) 


CENTRAL  FORCES,  21 3 
Multiplying  by  2du  and  integrating,  we  have 

From  the  initial  conditions  and  Eq.  (558)  we  have 

^=¥-1^' (589) 


and  therefore 


# 


5gi  +  «=^  +  -^-^ (S90) 


This  equation  may  be  written 


by  assuming 


^^  =  c'-{u-by (591) 


¥  =  ^    and    _-^_  +  ^  =  ...      .    .    (59.) 


From  Eq.  (591)  we  have,  taking  the  negative  sign  of  the  radical, 

■vrT§T^^  =  ''-' (593) 


and  by  integration, 


cos-^^y-  =  e  +  y, (594) 


in  which  y,  the  constant  of  integration,  is  the  initial  angle  which 


214  MECHANICS  OF  SOLIDS. 

the  prime  radius  vector  makes  with  the  fixed  line  of  reference. 
From  Eq.  {594)  we  get 

...  .    u=^b-]-c  cos  {O-i-y), (595) 

or 

''  =  l,  +  ccoHd  +  y)' (596) 

and  substituting  the  values  of  b  and  c, 


which  may  be  written 

"^  ^  'rJjT^       +  I  .  cos  (B-^y) 

Comparing   this   with   the   polar   equation  of   a   conic   section 
referred  to  the  focus  as  a  pole, 

'•-T+T^o70' (599) 

we  see  that  (598)  is  the  equation  of  a  conic  section  referred  to 
the  focus  as  a  pole,  in  which 

'^y     r/   +^> ('°^> 

0  =  (9  +  X (602) 


CENTRAL  FORCES.  21 5 


Therefore  the  orbit  will  be  an  ellipsCy  parabola  or  hyperbola  accord- 
ing as 

V'R  —  2)U'  <  o,     =0     or     >  o; 


that  is,  as 


r<|/^,     =|/?      or     >|/^.     .     . 


183.  To  determine  the  meaning  of  y  -~ ,  we  have 


(603) 


R 


1?  ~  ~~?' 


(604) 


Multiplying  by  2dr  and   integrating  between  the   limits  r  =  00 
and  r  =  jRy  we  have 

or 


.=/? 


(606) 


Thus  y  ^  is  shown  to  be  the  velocity  which  the  body  would 

have  if  it  should  move  from  rest  at  infinity  to  the  distance  J^ 
under  the  action  of  the  central  force;  it  is  called  l/ie  velocity  from 
infifiity  at  the  distance  R. 

Hence  we  conclude  that  the  orbit  of  a  body  under  the  action  of  a 
central  attraction  varying  inversely  as  the  square  of  the  distance  will 
be  an  ellipse,  parabola  or  hyperbola  according  as  the  i?iitial  velocity  is 
less  than,  equal  to  or  greater  than  the  velocity  from  infinity  at  the  ini- 
tial point. 


2l6  MECHANICS  OF  SOLIDS. 

184.    To  find  the  velocity  at  any  point  of  the  orbit.     For  the  ve- 
locity at  any  point  we  have,  Eq.  (581), 


^du 


and  hence  from  Eq.  (590)  we  have 


z/'  =  F''  +  2^'u  -  ^' (608) 


or 


z-'=F'  +  2A<'(i-^] (609) 


from  which  the  velocity  corresponding  to  any  radius  vector  r  can 
be  found. 

We  also  see  from  Eq.  (609)  that  the  velocity  at  any  point  of 
the  orbit  will  always  conform  to  that  which  characterizes  the 
particular  orbit  in  question;  that  is,  if  the  orbit  be  a  parabola, 
for  example,  the  velocity  at  any  point  whose  radius  vector  is  r 

2U' 

will  always  be  equal  to  —  at  that  point,  and,  similarly,  less  than 

-^  for  the  ellipse  and  greater  than  -^  for  the  hyperbola. 

185.   To  find  the  time  of  description  of  any  portion  of  the  orbit. 
(i)   The  Elliptical  Orbit. — We  have  for  the  equation  of  the  orbit 

I  I    +   <?   cos    d  r,         X 

-  =u=  — ^r-;      .....     (610) 

r  ayj  —  e)  ^      ^ 

hence 

du  e  sin  6  /^     \ 


CENTRAL  FORCES.  21/ 

and 

d^u  e  cos  d  tc     \ 


Therefore 

!(i  -  e') 


and  since 

A-  ^, 
we  have 

l^'=    t^'    ,x, (614) 

or 

//=   Vyw'^(i  -^») (615) 

Therefore  the  periodic  time  is 


y=     ,  =  2;ry -^ (616) 

We  also  have,  from  Eqs.  (551)  and  (615), 

dt  z= -dO  = ——=== (617) 

Differentiating  Eq.  (610),  and  substituting  for  sin  0  its  value  de- 
duced from  the  same  equation,  we  have 


-lft=       «^r^R^^_ 

r  f'aV  -{r  -ay  ^       ' 


21 8  MECHANICS  OF  SOLIDS. 

Substituting  in  Eq.  (617),  we  have 

,,  ^  JJ_  rdr  ^  J^  \{r  -  a) -^  a\d(r  -  a) 

^'  Va'e'  -(r-  af  M        VaV  -  (r  -  a)'  ^ 

Integrating  between  the  limits  corresponding  to  the  nearer  ver- 
tex r^  =  a(i  —  e)  and  any  value  of  r,  we  have 

t  =  \/^,  [a  sin -1  "^^^  -  V^V  -ir-  a)"]^ 

=  4/j  {a  cos-'  ^  -  V«V  -{a-  rf),    .     (620) 

from  which  the  time  of  description  of  any  portion   of  the  orbit 
can  be  found. 

Making  r  =  a(i  +  ^),  corresponding  to  the  farther  vertex,  we 
have  for  the  semi-periodic  time 

^=^1/? (6^1) 

Making  r  =  a,  corresponding  to  the  extremity  of  the  conju- 
gate axis,  we  have 


/ 


ii-^)^-' (^"> 


and  hence  for  the  time  from  the  extremity  of  the  conjugate  to 
the  farther  extremity  of  the  transverse  axis  we  have 


C-+')*^ 


,=  ^  +  AV-, (<i.3> 


CENTRAL  FORCES.  219 


From  these  values  we  see  that  the  Velocity  decreases  from  the 
nearer  to  the  farther  extremity  of  the  transverse  axis,  and  then 
increases  to  the  nearer  extremity. 

The  trajectory  of  a  projectile  in  vacuo,  under  the  supposi- 
tion that  its  weight  acts  with  constant  intensity  in  parallel 
directions,  was  shown  in  Art.  93  to  be  a  parabola.  The  pre- 
ceding discussion  shows  that  this  trajectory  will  be  an  arc  of  an 
ellipse  having  the  earth's  centre  at  the  farther  focus,  when  grav- 
ity is  considered  as  a  central  force  varying  inversely  as  the 
square  of  the  distance  from  the  centre  of  the  earth.  The  trans- 
verse axis  of  the  ellipse  is  the  vertical  through  the  highest  point 
of  the  trajectory. 

(2)  The  Parabolic  Orbit. — We  have  for  the  equation  of  the 
orbit 

r  =  -^—, (6^4) 

I  -j-  cos  u 

in  which  2a  is  the  semi-parameter.     We  therefore  have 

^  =  ^.^^  +.]=--  =  --,;     ,     .     .^    (625) 


hence 


and 


/*'  =  — ,     or    h"  =  2a^', (626) 


2a 


P 
We  have  from  Eq.  (551) 


^  =j»=n^ +^j=-7 (^^7) 


dt  =  -de\ 

n 


hence 


220  MECHANICS  OF  SOLIDS. 


~  ^  '^'M,   (i  +  cos  ey 

=  y  —j-i  tan  -»  —  tan  -^  +  -  tan'  -' tan'  -'  .     (629) 


n  n 

Place  tan  -"  =  /,  and  tan  -^  =  t^^  then  the  equation  above  be- 


comes 


=t/g«. ->,)(■ +'••+'•;■+'•) (630) 


Let/  =  I  4-  ^'^  ^    ,  and  we  have 

4 


Let  ^  be  the  length  of  the  chord  joining  the  extremities  of  the 
radii  vectores  r,  and  r^\  then  we  have 

^•  =  r.'-ir.r,cos(<?.-0.)  +  r,' 

=  (r,  cos  0,  —  r,  cos  8^)'  -\-  (r,  sin  ^,  —  r,  sin  S,)'.    (632) 


CENTRAL  FORCES.  221 

We  have  also,  since  r  =  r~^g  =  33^  =  «(■  +  tan»  ^(f), 

r,  =  a(i+0,     r,  =  a{x+i,y,       .     .     .     (633) 

I   —  f  "  2t 

COS  d^  =  YZfr^    ^^"  ^»  =  11^'''  *    '    *    ^^^"^^ 
^^^  ^>  =  fr^"    ^'"  ^*  "^  ^T^''*  *   '   '    ^^^^^ 

Therefore 

=  4a\t,-i,Y/; (636) 

hence 

c=2a(f,-ljy (637) 

and 

=  ^''\y+'-^'Yi (638) 

and  similarly 

r,  +  r,-c=2a^y-'  i^^  ^ (639) 

Substituting  in  Eq.  (631),  we  have  finally 

'  =  ^j  *'(v+^T^-'»^(v+^^^'};     (640) 

from  which  /  can  be  found  in  terms  of  the  radii  vectores  and  the 
chord  of  the  parabolic  arc. 


VM'^(e''  -  i) 

rdr 


222  MECHANICS  OF  SOLIDS. 

(3)  The  Hyperbolic  Orbit. — We  have  for  the  equation  of  the 
orbit 

r  =  — ^^ — 

I  4-  ^  cos  6^' 

or 

I  e 

U  =  — r^ r  +  --7-5 T  cos  ^;  .       .       .       .       (641) 

a(e^  —  I)       a(f  —  i)  v  -^  / 

and  therefore 

_     />«  r'^dd 

/^V^(.-i)4/{r  +  ay  -^V 

=  4/^  { 4/(.+.r-.v-.iog-+-+^(;+-)'i:£:f!  [ ;  (64.) 

from  which  the  time  corresponding  to  the  description  of  that 
part  of  the  orbit  from  the  vertex  to  the  point  corresponding  to 
any  radius  vector  r  can  be  found. 

186.  The  Anomalies. — When  the  central  attraction  varies  in- 
versely as  the  square  of  the  distance,  the  position  of  the  body  in 
its  orbit  is  generally  referred  to  the  right  line  coinciding  with 
the  least  radius  vector.  This  line  is  called  the  line  of  apsides ^  and 
its  intersections  with  the  orbit  are  called  apses;  the  one  nearer 
to  the  focus  being  the  lower^  and  the  other  the  higher^  apsis.  The 
angle  included  between  the  line  of  apsides  and  the  radius  vector 
is  called  the  anomaly^  and  is  measured  from  the  lower  apsis  as  an 
origin. 

Let  us  place 

cos-i— ^j-  =  «^, (643) 

an  auxiliary  angle;  then  we  have 

r  =  a{\  —  e  COS,  n) (644) 


CENTRAL  FORCES. 


223 


Substituting  the  value  of  u  in  Eq.  (620),  we  have,  after  placing 

nt  =  u  —  e  s\n  u (^45) 

Equating  the  values  of  r  from  Eqs.  (644)  and  (610),  we  have 

.     .     .     (646) 


1  —e' 


=z  I  —  e  cos  u, 


whence 


and  therefore 


or 


I  -{-  e  cos  d 

_  (i  —  e)(i  -fcos  u)^ 


I  +  cos  6 
I  —  cos  6 


1  —  e  cos  u 
(i  +  ^)  (i  —  cos  u) 


I  —  e  cos 


—  cos  6      I  4"  ^  I  —  cos  u 


I  -\-  cos  6       1—^*1  +  cos  «*' 


tan-^ 
2 


./i  +  e         I 
y  — ! —  tan  -u. 


(647) 


—  e 


(648) 
(649) 


The  angle  6  is  called  the  true  anomaly,  u  the  eccentric  anomaly^ 
and  ;«/  the  mean  ano??ialy. 

From  Eqs.  (644),  (645)  and  (649)  the  values  of  the  true 
anomaly  ^and  the  radius  vector  r  can  be  found  in  terms  of  the 
eccentricity  and  the  mean  anomaly.  (See  Price's  Calculus,  vol. 
iii.  pp.  561-567.)     These  are 


d=nt-\-2e  sin  nt-\-  —e*  sin  2nt-\ (13  sin  snt—^  sin  nt)-\-etc.; 


•=«(  I  — ^cos  nt-\ — (i  —cos  2nt) — -  (cos 3«/— cos  «/)-|-etc.  J. 


(650) 


224 


MECHANICS  OF  SOLIDS. 


Hence,  knowing  the  mean  motion,  the  time  since  the  epoch, 
and  the  eccentricity,  the  true  anomaly  or  the  angular  distance  of 
the  body  from  the  line  of  apsides,  and  the  distance  of  the  body 
from  the  focus  at  once  results.  The  difference  between  the  true 
and  mean  anomaly  d  —  7it,  called  the  Equation  of  the  Centre,  is 
evidently  a  function  of  the  eccentricity,  and  its  value  is  obtained 
at  any  time  /  from  the  first  of  Eqs.  (650). 

187.  To  illustrate  the  geometrical  meaning  of  these  quanti- 
ties let  APB'A^  Fig.  64,  be  the  elliptical  orbit,  .V  the  centre  of 
force  afthe  focus,  P  the  position  of  the  body,  SP  ~  r,  PSA  = 
6  and  AC  =  a.  On  AA^  describe  a  semicircle  APA\  From 
the  properties  of  the  ellipse  we  have 


SP  =  a-  eCM, 


and  therefore 


r  •=^a  —  ae  cos  QCM 
=  «  —  ae  cos  u,     . 


(651) 
(652) 


From  Eq.  (616)  we  see  that  the  periodic  time  is  independent 
of  the  eccentricity  of  the  ellipse;  it  is  therefore  the  same  as  that 
in  the  circle  whose  radius  is  a;  but  in  this  case  e  =  o,  r  =  a,  6  =1 

u  =  nt.  Hence  nt  represents  the  arc  of 
the  circle  which  would  be  described 
uniformly  by  a  body  in  the  same  time 
as  that  in  which  the  elliptic  arc  is  de- 
scribed, both  bodies  starting  from  A^ 
and  both  reaching^'  at  the  same  time; 
71  is  therefore  called  the  mean  motion 
of  the  body.  Since  sin  u  is  positive  in 
the  first  two  quadrants,  we  see,  Eq. 
(645),  that  u  is  greater  than  nt  while 
the  body  is  describing  that  part  of  its 
orbit  from  A  to  A\  and  less  than  nt  from  A'  to  A;  therefore  the 
true  place  of  the  body  is  in  advance  of  its  mean  place  in  the  first 
and  second  quadrants,  and  behind  in  the  third  and  fourth;  nt  is 


Fig.  64. 


THE   SOLAR  SYSTEM.  22$ 


therefore  called  the  mean  anomaly,  and  since  u  depends  on  the 
value  of  ^,  «*  is  called  the  eccentric  anomaly.  Both  the  velocity 
and  the  angular  velocity  are  greatest  at  A  and  least  at  A\  as  is 
seen  from  the  equations  giving  these  values  in  the  laws  of  cen- 
tral forces. 


The  Solar  System. 

l88.  The  Solar  System  consists  of  the  sun  and  other  bodies 
whose  relative  positions  and  motions  are  mutually  dependent, 
and  which  taken  together  may  be  considered  as  a  single  system 
of  bodies  in  space.  It  derives  its  name  from  the  sun,  the  great 
central  body  about  which  all  the  other  members  of  the  system, 
called /r/Vz/^ry  or  secondary  bodies,  revolve. 

Th.Q primary  bodies  are: 

(i)  The  four  inner  or  lesser  planets.  Mercury,  Venus,  the  Earth 
and  Mars,  named  in  order  of  their  distance  from  the  sun. 

(2)  A  group  of  minor  planets  called  Asteroids,  of  which  over 
two  hundred  and  sixty  have  so  far  been  noted  and  catalogued. 

(3)  The  four  outer  or  greater  planets,  Jupiter,  Saturn,  Uranus, 
and  Neptune. 

(4)  A  number  of  Comets  and  Meteors,  or  bodies  having  masses 
much  smaller,  and  generally  orbits  of  much  greater  eccentricity, 
than  those  of  the  planets  and  asteroids. 

The  secondary  bodies  are  the  Satellites  or  Moons  of  the  planets, 
which  describe  orbits  about  the  latter  and  are  carried  with  them 
in  orbital  motion  about  the  sun;  of  these  now  known  three  be- 
long to  the  inner  and  seventeen  to  the  outer  planets. 

All  the  bodies  of  the  solar  system  are  spheroidal  in  form,  and 
their  diameters  are  very  small  compared  with  the  distances 
which  separate  them  from  each  other.  In  addition  to  their  or- 
bital motions  they  have  a  motion  of  rotation  about  their  axes. 

The  mass  of  the  sun  is  more. than  seven  hundred  and  forty 
times  as  great  as  the  sum  of  the  masses  of  all  the  other  bodies 
of  the  system.  Owing  to  this  fact,  and  to  the  relative  positions. 
15 


226  MECHANICS  OF  SOLIDS. 

of  the  planets,  the  centre  of  mass  of  the  entire  system  lies  within 
the  sun's  volume  and  not  far  from  its  own  centre. 

189.  Kepler  s  Laws. — John  Kepler,  of  Wurtemburg,  was  the 
first  to  announce  the  laws  governing  the  motion  of  the  planets 
about  the  sun.  This  announcement  was  the  result  of  more  than 
twenty  years'  faithful  and  laborious  study  of  the  observations 
collected  by  his  predecessor  Tycho  Brahe.  Kepler's  investiga- 
tions were  principally  directed  to  the  explanation  of  the  appar- 
ent irregularities  of  the  motion  of  the  planet  Mars,  whose  orbit 
was  at  that  time  supposed  to  be  an  epicycloid.  These  laws  of 
Kepler  not  only  completely  accounted  for  the  motion  of  Mars,  but 
also  satisfactorily  explained  the  motions  of  all  the  other  planets 
about  the  sun,  and  those  of  the  satellites  about  their  respective 
primaries.     These  laws  are: 

(i)  The  orbit  of  each  planet  about  the  sun  is  an  ellipse^  having  one 
of  its  foci  in  the  suits  centre. 

(2)  The  areas  described  by  the  radius  vector  of  each  planet  in  its 
orbital  motion  vary  directly  as  the  times  of  describing  them. 

(3)  The  squares  of  the  periodic  times  of  the  planets  are  directly 
proportional  to  the  cubes  of  their  fnean  distances  from  the  suns  centre. 

190.  If  we  assume,  for  tlie  present,  that  these  laws  are  accu- 
rately true,  we  readily  deduce  the  following  consequences,  viz.: 

(i)  The  orbit  of  each  planet  being  an  ellipse  having  the  sun's 
centre  in  one  focus,  it  follows  that  the  value  of  the  relative  ac- 
celeration becomes,  Eq.  (613), 


^  _       hW       _        h'         I 


that  is,  the  relative  acceleration  varies  inversely  as  the  square  of  the 
distance  of  the  planet  from  the  centre  of  the  sun. 

(2)  From  the  second  law,  or  that  of  equal  areas  described  in 
equal  times,  we  have 

r^dd  =  hdt  =  xdy  —  ydx (654) 


THE  SOLAR  SYSTEM.  22/ 

Whence,  after  differentiating  and  dividing  by  <//',  we  have 

Multiplying  by  M\  the  mass  of  the  planet,  we  have 

M-'^-^,x-M'^-^y=Yx-Xy=o.  .     .     .     (656) 

Therefore  the  action-line  of  the  reciprocal  attraction  of  the  sun  and 
the  planet  always  passes  through  the  sun's  centre;  and  since  this  re- 
ciprocal attraction  varies  inversely  as  the  square  of  the  distance, 
the  force  which  keeps  the  planet  in  its  orbit  is  a  cejitral  force. 

(3)  From  Eq.  (6i6)  we  have  for  the  square  of  the  periodic 
time  of  one  of  the  planets  whose  mean  distance  is  a*  and  whose 
mass  is  M' 

r^  =  '-^; (657) 


in  which  /i'  is  the  intensity  of  the  central  attraction  for  a  unit 
mass  at  a  unit's  distance.  Similarly  for  another  planet  whose 
distance  is  a"  and  mass  J/"  we  have 


A^tfi 


whence  we  have 


^»  =  ^; (658) 


From  Kepler's  third  law,  we  have 

T'^  ~  a' 


(660) 


228  MECHANICS  OF  SOLIDS, 

which  being  substituted  in  the  preceding  equation  gives 

That  is,  the  rigid  truth  of  Kepler's  third  law  involves  the 
equality  of  the  central  attraction  for  all  the  planets.     But 

/<'  =  (Jf  +  M')^     and     /t"  =  {M -\-  M'')/i;     .     (66 1> 

hence,  since  the  masses  of  the  planets  are  known  to  be  unequal, 
we  must  conclude  that  Kepler's  third  law  is  not  rigidly  true. 

If  the  masses  of  the  sun,  1,000,000,000,  of  Jupiter,  954,305,  the 
greatest,  and  of  Mercury,  200,  the  least  of  the  planets,  be  substi- 
tuted in  (661),  we  find 

^  =  1.000954, (662) 

and  this  ratio  will  be  more  nearly  equal  to  unity  for  any  other 
pair  of  planets.     The  discrepancy  in  assuming  unity  for  the  ratia 

^  for  the  planets  of   the  solar  system    is   therefore   in   general 

negligible,  and  the  consequence  of  Kepler's  third  law  may  be 
taken  as  true  within  sensible  limits. 

191.  -Law  of  Gravitation. — Later  and  more  accurate  observa- 
tions than  those  which  Kepler  employed  show  that  his  laws  are 
not  exactly  true,  but  are  only  very  close  approximations  to  the 
truth.  The  single  law  which  governs  planetary  motions  and 
definitely  fixes  their  actual  departures  from  the  positions  as- 
signed by  Kepler's  laws  is  that  of  universal  gravitation,  which 
is  thus  enunciated  by  Isaac  Newton: 

That  every  particle  of  matter  in  the  universe  attracts  every  other 
particle^  with  an  intensity  which  varies  directly  as  the  product  of  their 
masses^  and  ifiversely  as  the  square  of  the  distance  which  separates  them. 

Newton  deduced  this  law  from  his  investigations  of  the  rela- 
tive acceleration  of  the  moon,  in  a  direction  normal  to  its  orbit 


THE   SOLAR  SYSTEM.  229 

about  the  earth.  He  proved  tliat  the  earth's  relative  attraction 
on  the  moon  caused  it  to  fall  towards  the  earth,  with  an  acceler- 
ation due  to  gravity,  modified  only  by  the  increased  distance  and 
greater  mass  of  the  moon,  precisely  as  a  body  near  the  earth's 
surface  falls  with  its  particular  acceleration;  and  therefore  con- 
cluded that  the  law  of  attraction  between  the  earth  and  moon 
was  essentially  the  same  as  that  between  the  earth  and  body. 
From  this  deduction  the  generalization  to  the  enunciated  law  of 
gravitation  followed. 

The  intensity  of  the  reciprocal  attraction  between  the  sun 
and  a  planet,  whose  masses  are  M  and  M'  respectively,  under 
the  law  of  gravitation  is  therefore 

G  =  -—t-m; (663) 

/f  in  this  expression  being  the  intensity  of  the  reciprocal  attrac- 
tion of  a  unit  mass  for  another  unit  mass  at  a  unit's  distance. 
Therefore  the  partial  differential  equations  of  the  undisturbed 
orbit  of  a  planet  about  the  sun,  Eqs.  (546),  become,  under  the  law 
of  gravitation, 

d'x  MM' 

-}XX\ 


dt""  r 

d\  MM' 

d^'z  _  MM' 

dt'  -  -7^^^' 


(664) 


Kepler's  first  and  second  laws  can  be  deduced  directly  from 
these  equations,  and  hence  are  simply  the  consequences  derived 
from  the  undisturbed  orbit  of  a  primary  about  the  sun  under  the 
supposition  that  the  law  of  gravitation  is  the  governing  law  of 
their  inutual  attraction. 

The  differential  equations  of  the  actual  or  disturbed  orbit  can 
be  obtained  immediately  from  Eqs.  (545),  by  substituting  the 
values  which  the  symbols   (MM'),   {MM")   and  (M'M")  take 


230  MECHANICS  OF  SOLIDS. 

under  the  law  of  gravitation,  and  the  resulting  equations  will 
differ  from  (664)  only  in  the  third  and  fourth  terms.  The  latter 
being  computed  and  applied  properly  to  the  undisturbed  orbit 
will  give  the  actual  orbit  of  the  planet.  These  terms,  called  the 
perturb ating  functions  of  the  orbit,  depend  upon  the  relative  attrac- 
tions of  the  other  planets  for  tlie  sun  and  for  the  planet  whose 
orbit  is  to  be  determined.  Owing  to  the  relatively  great  mass 
of  the  sun,  and  to  the  immense  distances  which  separate  the 
planets  from  each  other,  \\i^ perturbations  or  actual  displacements 
of  a  planet  from  its  undisturbed  orbit  are  sensibly  infinitesimal 
quantities,  compared  with  the  actual  distance  of  the  planet  from 
the  sun.  Hence  the  component  rectangular  displacements  due 
to  the  perturbating  action  of  each  planet  may  be  computed  for 
each  planet  separately  as  if  it  alone  acted;  then  the  algebraic  sum 
of  the  separate  perturbations  in  any  direction  may  be  taken  as 
the  resultant  perturbating  effect  in  that  direction  due  to  the 
simultaneous  action  of  all  the  planets,  without  the  least  appreci- 
able  error.  Because  of  this  fact,  tlie  problem  is  called  the  prob- 
lem of  three  bodies^  viz.,  the  sun,  the  planet  and  the  perturbating 
body. 

The  theoretical  deductions  which  flow  from  the  assump- 
tion of  the  law  of  universal  gravitation  as  the  governing  law  of 
planetary  motion  have  been  amply  confirmed  by  the  accurate 
astronomical  predictions  of  the  positions  and  motions  of  the 
planetary  bodies,  made  years  in  advance,  and  markedly  so,  by 
the  circumstances  attending  the  discovery  of  the  planet  Neptune; 
so  that  the  law  itself  is  at  present  accepted  as  the  fundamental 
law  of  physical  astronomy. 

192.  Planetary  Orbits. — The  undisturbed  orbit  of  each  of  the 
bodies  of  the  solar  system  has  been  shown  to  be  a  plane  curve, 
whose  plane  passes  through  the  sun's  centre;  but  it  is  ascer« 
tained,  by  observation,  that  no  two  of  these  planes  are  coincident. 
In  order  to  find  the  relative  positions  and  motions  of  the  bodies 
of  the  solar  system  at  any  time,  it  is  necessary  to  refer  them  to 
the  surface  of  the  celestial  sphere  by  some  system  of  spherical 
co-ordinates. 


THE   SOLAR   SYSTEM.  23 1 

The  celestial  sphere  is  usually  taken  to  be  that  sphere  which 
is  enclosed  by  the  surface  of  the  visible  heavens;  but  it  may  be 
taken  to  be  any  sphere  whose  centre  is  tlie  position  of  the  ob- 
server, and  whose  radius  is  entirely  arbitrary.  The  Ecliptic  is  the 
great  circle  of  intersection  of  the  celestial  sphere,  by  the  plane 
of  the  earth's  orbit.  The  Celestial  Equator,  or  Equinoctial,  is  the 
great  circle  of  intersection  of  the  celestial  sphere,  by  the  plane  of 
tiie  earth's  equator.  The  poles  of  the  heavens  are  the  poles  of 
the  Equinoctial,  or  are  the  points  in  which  the  earth's  axis  pro- 
duced pierces  the  celestial  spiiere.  T ho.  Equinoxes  are  the  points 
in  which  the  equinoctial  and  ecliptic  intersect;  the  Vernal  Equi- 
nox being  that  point  in  which  the  sun  appears  at  the  beginning 
of  spring,  and  the  Autu?nnal  Equinox  that  in  which  it  appears  in 
the  beginning  of  autumn.  Celestial  longitude  and  latitude  are 
spherical  co-ordinates  by  which  any  point  is  referred  to  the  plane 
of  the  Ecliptic,  and  to  that  of  a  great  circle  of  the  celestial  sphere 
perpendicular  to  the  ecliptic,  passing  through  the  vernal  equi- 
nox. Celestial  longitude  is  the  angular  distance  from  the  vernal 
equinox,  measured  on  the  ecliptic  eastward ly  in  direction,  to  the 
circle  of  latitude  which  passes  through  the  position  in  question; 
-eiud  celestial  latitude  \s  the  angular  distance  to  the  given  point, 
from  the  ecliptic,  measured  on  that  circle  of  latitude  which 
passes  through  the  given  point.  The  line  of  intersection  of  the 
plane  of  a  planet's  orbit  with  the  plane  of  the  ecliptic  is  called 
\.\\Qline  of  nodes;  the  ascending  node  being  the  point  of  the  planet's 
orbit  at  which  the  planet  passes  from  south  to  north  of  the 
ecliptic,  the  other  being  the  descending  node.  The  nearest  point 
of  a  planet's  orbit  to  the  sun  is  C3.\\&6.  perihelion,  and  the  farthest 
is  called  aphelion. 

The  elements  of  a planefs  orbit  are  seven  in  number,  viz.: 
(i)  The  inclination  of  its  plane  to  the  plane  of  the  ecliptic. 

(2)  The  longitude  of  the  ascejiding  node. 

(3)  The  orbit  longitude  oi  perihelion. 

(4)  The  mean  distance  of  the  planet  from  the  sun,  or  the  semi- 
transverse  axis  of  the  planet's  orbit. 

(5)  The  eccentricity  of  the  orbit. 


232  MECHANICS  OF  SOLIDS. 

(6)  T\\Q  position  of  the  planet  at  any  given  time,  as  at  \.\\q  epoch. 

(7)  The  ?}ica?i  orbital  fnotion. 

The  first  two  of  these  elements  fix  the  plane  of  the  orbit  with 
reference  to  the  plane  of  the  ecliptic;  the  third  fixes  the  position 
of  the  line  of  apsides  in  this  plane  and  from  which  the  anomalies 
are  reckoned;  from  the  fourth  and  fifth  the  form  and  dimensions 
of  the  ellipse  are  determined;  and  the  last  two,  called  the  ele- 
ments of  position,  locate  the  planet  in  its  orbit  at  any  time. 

The  elements  of  any  planetary  orbit  are  deduced  from  three 
consecutive  observations  of  its  position  in  right  ascension  and 
declination  and  the  times  of  observation,  by  methods  which  will 
be  explained  in  the  course  in  Astronomy. 

Since  the  mean  motion  depends  on  the  periodic  time,  and  the 
latter,  by  Eq.  (616),  depends  on  the  mass  of  the  planet,  it  is  nec- 
essary to  explan  how  the  mass  of  a  planet  is  ascertained.  The 
masses  of  the  planets  are  so  small,  in  comparison  with  the  mass 
of  the  sun,  that  their  values  cannot  be  ascertained  from  Eq. 
(657)  after  substituting  the  observed  periodic  times  and  mean 
distances.  But  if  we  consider  the  planet  Jupiter  and  one  of  its 
satellites,  we  will  have,  for  the  periodic  time  T'  of  the  latter 
about  Jupiter, 

^'  =  ^T-.^ (^^3) 

in  which  a'  is  its  mean  distance  from  Jupiter,  and  m  its  mass;  /(, 
the  attraction  of  a  unit  mass  at  a  unit's  distance,  being  here 
taken  as  unity.  Similarly,  for  the  periodic  time  of  Jupiter  about 
the  sun,  we  have 


whence  we  have 


THE  SOLAR  SYSTEM,  233 

if  ;;/  be  supposed  small  compared  with  M\  and  M'  small  com- 
pared with  M.  Substituting  the  known  values  of  T^  T',  a  and 
^',  we  have  the  ratio  of  the  mass  of  Jupiter  to  that  of  the  sun. 
In  the  same  way  the  masses  of  all  the  planets  having  satellites 
may  be  compared  with  that  of  the  sun;  and  if  the  mass  of  one  of 
these  be  found,  that  of  the  remaining  planets  will  at  once  result. 
To  find  the  mass  of  the  planets  Mercury  and  Venus^  which  have 
no  satellites,  recourse  must  be  had  to  their  perturbating  effects 
on  the  other  planets. 

The  mass  of  the  earth  has  been  ascertained  by  direct  meas- 
urement of  its  figure,  magnitude  and  density.  From  the  direct 
geodesic  measurement  of  the  arcs  of  the  meridian  in  England, 
France,  Russia,  India  and  Africa,  the  form  and  dimensions  of 
the  earth  have  been  determined.  The  density  has  been  directly 
investigated  by  means  of  Dr.  Maskelyne's  observations  with  the 
pendulum  near  Schehallien  Mountain  in  Scotland;  and  also  by 
the  experiments  of  Cavendish  and  Bailey  from  the  attraction  of 
leaden  balls  on  small  masses.  From  these,  the  mass  of  the  earth 
having  been  found,  the  masses  of  the  sun  and  the  other  planets 
are  readily  obtained.  The  further  discussion  of  planetary  mo- 
tions is  reserved  for  the  course  in  Astronomy. 


THEORY  OF  MACHINES, 


193.  Thus  far  we  have  regarded  bodies  as  rigid  solids,  eitlier 
wholly  or  partially  free  to  move  under  the  action  of  extraneous 
forces.  But  the  various  devices  or  machines  designed  for  the 
transfer  of  energy  from  one  system  of  masses  to  another  are 
made  up  of  parts  which  are  neither  free  nor  rigid;  and  in  their 
use  certain  resistances  are  developed  by  the  active  extraneous 
forces,  which  malce  the  actual  results  differ  from  the  theoretical 
more  or  less  widely.  It  is  the  office  of  Experiment  to  find  the 
values  of  these  resistances,  to  tabulate  the  results,  and  to  deduce 
therefrom  the  experimental  Imvs  covering  all  cases  that  may  arise 
in  practice.  We  will  consider  briefly  the  resistances  of  Friction 
and  Stiffness  of  Cordage. 

Friction. 

194.  Friction  is  the  resistance  which  the  surface  of  one  body 
offers  to  the  sliding  or  rolling  upon  it  of  any  other  body.  It  is 
due  to  the  roughness  of  the  surfaces  of  bodies;  for  as  no  degree 
of  polish  can  make  any  surface  perfectly  smooth,  there  will 
always  be  minute  projections  on  one  surface  which  interlock 
with  those  of  the  other.  These  projections  must  be  broken 
down,  abraded  or  lifted  over  each  other  before  motion  can  take 
place.  When  the  roughness  of  any  two  surfaces  is  diminished 
by  polish  or  lubricants  the  friction  between  them  decreases. 

The  friction  which  opposes  a  change  of  the  body  from  rest 
to  motion  is  called  static  friction,  and  that  which  accompanies 
motion  is  called  kinetic  friction.  The  latter  may  be  either  sliding 
or  rolling;  thiis  a  heavy  body  dragged  over  a  surface,  an  axle 


FRICTION.  235 


turning  in  a  journal-box,  and  a  vertical  shaft  turning  on  a  hori- 
zontal plane,  give  examples  of  sliding  friction;  while  a  wheel 
rolling  over  the  surface  of  the  ground  is  resisted  by  rolling  fric- 
tion. Sliding  friction,  being  the  more  common,  will  be  alone 
discussed. 

The  action-line  of  friction  coincides  with  the  tangent  to  the 
surfaces  at  the  point  of  contact,  and  its  direction  is  always  op- 
posite to  that  of  the  motion.  The  intensity  of  friction  must  al- 
ways be  determined  by  experiment,  or  may  be  assumed  trom 
previous  experiments  on  similar  bodies  and  surfaces.  Such 
results  iiave  been  generalized  into  what  are  known  as  the  laws 
of  friction.  The  accepted  laws  have  been  deduced  from  the  ex- 
periments made  by  Coulomb  in  1781,  and  from  those  made  by 
Morin  under  the  direction  of  the  French  Government  in  1830-4. 
They  are  : 

(i)  The  intensity  of  friction^  for  the  same  material  surfaces,  varies 
directly  with  the  normal  pressure. 

(2)  The  intensity  of  friction  is  independent  of  the  area  of  contact  of 
the  surfaces. 

(3)  The  intensity  of  friction  is  independent  of  the  velocity  of  motion 
of  the  rubbing  surfaces. 

Recent  experiments  indicate  that  the  last  law  is  only  ap- 
proximately true  for  velocities  less  and  greater  than  those  em- 
ployed in  the  experiments  of  Morin;  and  since  the  laws  are 
wholly  experimental,  we  are  warranted  in  accepting  them  only 
within  the  limits  covered  by  the  experiments  from  which  they 
were  deduced.  In  static  friction  there  are  also  variations  de- 
pending on  the  length  of  time  during  which  the  surfaces  have 
been  in  contact,  and  therefore  greater  discrepancies  occur  in  this 
kind  than  in  kinetic  friction. 

195.  Coefficient  of  Friction. — If  iV  represent  the  normal  pres- 
sure, F  the  intensity  of  friction  for  any  two  surfaces  due  to  Ny 
and /"the  intensity  of  friction  for  a  unit  of  normal  pressuie,  we 
have,  from  the  first  law, 


F=fN,     or    /=^. (668) 


236  THEORY   OF  MACHINES. 

f  is  called  the  coefficient  of  friction,  and  when  known  for  any  two 
surfaces  the  total  friction  for  those  surfaces  can  be  found  for  any 
normal  pressure.  Its  value  depends  upon  the  nature  of  the  rub- 
bing surfaces,  upon  their  smoothness,  and  upon  the  degree  of 
lubrication  given  lo  them. 

To  find /experimentally  in  any  particular  case,  let  the  body 
be  placed  upon  a  plane  surface,  Fig.  65,  and  let  the  latter  be 
gradually  inclined  to  the  horizon   until  the  body  is  in  a  state 

bordering  on   motion  ;  then  if  motion 
be  given   the  body,  it  will   descend  the 
^ws$7m  plane   uniformly.     The    forces  are  the 

weight  of  the  body  acting  vertically 
downward,  and  the  friction  resisting  up 
the  plane.  The  normal  component  of 
the  weight,  if  a  be  the  angle  made  by  the 
^^^-  ^5-  plane  with  the  horizon,  is  ^cos  a,  and 

hence  the  friction  \s  fW  cos  a.  The  component  of  the  weight 
parallel  to  the  plane  urges  the  body  down  the  plane,  and  is 
W  sin  a\  and  since  the  motion  is  uniform  the  intensity  of  the 
parallel  component  must  be  equal  to  that  of  the  resistance  due 
to  friction.     Hence  we  have 

f  IV  cos  a  =  JV  sin  a, (669) 

and  therefore 

/=  tan  a (670) 

That  is,  t/ie  coefficient  of  friction  for  any  two  materials  is  equal  to 
ihe  natural  tangent  of  the  inclination  to  the  horizon  which  a  plane  of 
one  of  the  substances  must  make  in  order  that  a  body  of  the  other  sub- 
stance may  descend  uniformly  due  to  its  weight  when  resisted  only  by 
friction. 

The  inclination  of  the  plane  is  usually  called  the  attgle  of  fric- 
iion^  and  sometimes  the  limiting  angle  of  resistance.     To  explain 
the  meaning  of  this  latter  term,  let  a  body  rest  on  a  plane  in-* 
clined  at  the  angle  of  friction;  it  is  then  said  to  be  in  a  state 


FRICTION.  237 


bordering  on  motion  downward.  The  angle  at  the  centre  of 
gravity  of  tlie  body,  made  by  the  direction  of  the  weight  and  the 
normal  to  the  surface,  is  then  equal  to  or,  the  angle  of  friction. 
If  any  force  P  whose  action-line  lies  ivithin  the  angle  a  be  supposed 
introduced  at  the  centre  of  gravity,  it  is  evident  that  the  addi- 
tional developed  friction  arising  from  the  normal  component  of 
P  is  greater  than  the  component  of  P  parallel  to  the  plane; 
therefore  the  body  will  be  no  longer  in  a  state  bordering  on 
motion.  If  the  action-line  of  a  force  fall  outside  of  the  same 
angle  on  the  side  of  the  weight,  the  additional  friction  developed 
by  its  normal  component  will  be  less  than  the  intensity  of  its 
parallel  component ;  therefore  the  body  will  move  down  the 
plane  with  accelerated  motion.  If  the  action-line  of  any  force 
coincide  with  that  of  the  weight,  the  body  will  still  be  in  a  state 
bordering  on  motion  downward,  and  it  put  in  motion  it  will 
descend  the  plane  uniformly. 

196.  Problems  involving  Frictioti. — (i)  Motion  on  a  Plane  Surface^ 
Let  a  body  resting  on  a  plane.  Fig.  dd,  be 
acted  on  by  its  weight,  W^  and  let  any  ex- 
traneous force  whose  action-line  lies  in  a 
vertical  plane,  and  whose  intensity  is  /,  be 
applied  to  it;  let  /  be  the  inclination  of  the 
plane  to  the  horizon,  and  d  the  angle  made 
by  the  force  with  the  plane.     If  the  body  ^'^-  ^^• 

be  bordering  on  motion  downward  the  friction  will  act  up  the 
plane,  and  we  shall  have,  from  the  principles  of  equilibrium, 

/( J^cos /— /sin  6^)  =  fFsin  /  — /cos  ^,  .     .     (671) 
whence 

/=   ^^sin/-/cosj; 

If  the  body  be  bordering  on  motion  up  the  plane,  the  action- 
line  of  friction  will  change  direction  by  180°,  and  the  value  of  / 
can  be  obtained  from  the  preceding  by  changing  the  sign  of/;  or 


sin/+/cos/ 


238 


THEORY  OF  MACHINES. 


Therefore  the  value  of  /  may  vary  between  these  limits  without 
causing  motion  in  the  body. 

Considering  /  a  function  of  6^,  we  have,  after  differentiating 
the  last  equation, 


dl  .  .     .  .    J.         .V   sin  0  —  f  cos  d  tr     \ 

-:Ja  =  ^(sin  ^  +/C0S  t)-, a  ,    r   .     n-.^  •     •     (674) 

dd  ^  '  -^  ^(cos  ^+/sm^)^  ^  '   '' 


Applying  the  condition  for  maxima  and  minima,  by  placing 
the  second  member  equal  to  zero,  we  find 


/  =  tan  6,     or     6  =  tan  -  i/, 


(675) 


which  corresponds  to  a  minimum  value  of  /;  therefore  a  force  is 
applied  to  the  best  advantage  in  moving  a  body  up  a  plane  when 
its  action-line  makes  an  angle  with  the  plane  equal  to  the  angle 
of  friction.  As  this  result  is  independent  of  /,  it  is  true  what- 
ever be  the  inclination  of  the  plane  to  the  horizon,  and  is  there- 
fore true  if  the  plane  be  horizontal. 

(2)  To  find  the  Friction  on  a  Trunnion. — The  cylindrical  pro- 
jections at  the  extremities  of  an  axle  are  called  trunnions;  the 
cylindrical  box  upon  which  a  trunnion  is  supported  is  called  a 
trunnion-bed  or  pillow-block.  When  the  axle  is  supported  on  its 
end,  the  latter  is  called  2,  pivot. 

Let  A^  Fig.  67,  be  the  trunnion,  B  the  trunnion-bed,  and  C 
any  element  of  contact  during  rotation;  let  R 
be  the  resultant  of  all  the  extraneous  forces 
acting  on  the  trunnion,  excluding  friction;  N 
and  T  the  normal  and  tangential  components 
of  R  respectively.  If  the  trunnion  rotate,  it 
will  rise  in  its  box  until  the  developed  friction 
F  is  equal  to  7",  after  which  sliding  will  occur 
at  the  element  of  contact.  Then  the  resultant 
of  R  and  F  will  be  normal  to  the  surface  of  the  trunnion  at  the 


\  i 

> 
A 

\, 

iM 

1; 

^ 

s-> 

^      B 

'\1 

FRICTION.  239 


element  of  contact,  and  will  be  equal  to  N.     Let  a  be  the  angle 
between  R  and  N\  then  we  have 

N=RcQsa     and     T  =  R  sia  a  =1  F\     .     .     (676) 
whence 

_=/=tanaf (677) 

Therefore  the  element  of  contact  during  rotation  is  that  at  which 
the  normal  to  the  trunnion  makes  an  angle  with  the  resultant 
equal  to  the  angle  of  friction. 
To  find  the  friction  we  have 


(678) 


I  +  tan'' a       1+/"'     * 
multiplying  by/'^'  and  extracting  the  square  root,  we  have 

Fr=fN=fR  cos  a  =/R  '._.  =  R-j^-,  (679) 

y  I  +  tan"  a  Vi  +/" 

That  is,  the  friction  on  trunnions  is  equal  to  the  resultant  of  the 
extraneous  forces  multiplied  bv  ,  in  which  /  is  the  co- 

efficient  of  friction  for  the  materials  which  compose  the  trunnion 
and  box. 

The  moment  of  friction  on   trunnions,  if  r  be  the  radius  of 
the  trunnion,  is 

Rr   /       , (680) 

and  the  work  consumed  by  friction  in  n  complete  turns  is 

R27trn—-£=^. (681) 


240  THEORY  OF  MACHINES. 

From  the  second  law  of  friction  we  see  that  the  intensity  of 
friction  and  the  work  are  independent  of  the  length  of  the  trun- 
nion. 

(3)  Friction  on  a  Circular  Pivot. — Let  the  shaft  be  vertical  and 
its  pivot  end  a  circle  of  radius  a^  Fig.  68,  resting  on  a 
horizontal  surface;  and  let  the  centre  of  the  circle  of 
contact  be  the  origin  of  the  polar  co-ordinates  r  and  6^, 
which  fix  the  position  of  any  elementary  area  of  contact. 
Fig  68     Let/  be  the  intensity  of  the  normal  pressure  on  each 
unit  of  area,  which  is  equal  to  the  whole  normal  pres- 
sure divided  by  the  area  of  tiie  pivot  surface. 
The  expression  for  any  elementary  area  is 

rdrdd', 
the  normal  pressure  upon  it  is 

prdrdQ\ 
the  developed  friction  is 

fprdrdB', 

and  the  moment  of  this  friction  with  respect  to  the  axis  of  the 
shaft  is 

fpr'^drdd. 

Integrating  this  last  expression  between  proper  limits,  we  have 
for  the  resultant  moment  of  the  friction  on  a  pivot 

M=    I       I    fpr'drdd  =  %fp7ta\      .     .     .     (682) 

If  iV  represent  the  whole  normal  pressure,  we  have 

N=p7ta\ {eZz} 

whence 

M^fN^a (684) 


FRICTION.  241 


Therefore  the  moment  of  friction  on  a  pivot  is  equal  to  the  pro- 
duct of  the  coefficient  of  friction  for  the  materials  of  which  the 
shaft  and  support  are  made,  the  total  normal  pressure,  and  two 
thirds  of  the  radius  of  the  pivot  surface. 

From  this  we  see  that  the  moment  of  friction  may  be  dimin- 
ished by  decreasing  the  circular  area  of  the  pivot,  provided  the 
diminished  area  be  sufficient  to  withstand  the  normal  pressure 
without  penetrating  the  surface  on  which  it  rests.  The  distance 
\a  is  called  the  viean  lever  of  friction  on  pivots,  and  we  see  that 
if  the  whole  friction  be  supposed  concentrated  on  an  arc  of  this 
radius  its  moment  will  be  the  same  as  the  resultant  moment  of 
all  the  elementary  frictions. 

The  work  consumed  by  friction  in  11  revolutions  is 

27tnfNla. (685) 

(4)  Friction  on  a  Ring  Pivot. — Let  the  inner  and  outer  radii  of 
the  ring  be  a  and  a'  respectively;  then  (Eqs.  (682)  and  (683)) 

M=\fpn{a"  -a') (686) 

N=pn{a"-ay, (687) 


whence 


^=t/^S^ (^^*) 


If  b  be  the  breadth  of  the  ring  and  rbe  its  mean  radius,  we  have 

^'  =  r  +  \b, 
a  =  r  —  \b\ 

which  substituted  in  Eq.  (688)  give 

^=/A-(r+^g.     .....     (689) 

r6 


242 


THEORY  OF  MACHINES. 


The  factor 


Ki^f) 


is  called  the  mean  lever  of  friction  for 


a  ring  pivot. 

The  quantity  of  work  consumed  by  friction  on  pivots  in  n 
revolutions  of  the  shaft,  /  being  the  mean  lever  of  friction,  is 

27tnfNl. (690) 

(5)  Friction  of  a  Cord  on  a  Cylinder. — Let  R  be  the  radius  of 
the  cylinder,  s  the  length  of  the  cord  in  contact,  and  7\  and  T^ 
the  tensions  at  first  and  last  points  of  contact,  T^  being  greater 
than  T^.  If  there  were  no  friction  T^  would  be  equal  to  T^\ 
hence  the  excess  T^  —  T^,  when  the  cord  is  in  a  state  bordering 
on  motion,  is  due  to  friction.  Let  T  be  the  tensions  at  the  ex- 
tremities of  the  elementary  arc  ds^  and  let  6  be  the  angle  included 
between  their  action-lines.     Their  resultant  N  is  then 


JV=  VT''-\-2TTcose-\-T''  =  TV2{i  +  COS  6)  =  2  T  cos  id.  {691) 

If  0,  Fig.  69,  be  the  angle  at  the  centre  of 
the  cylinder  subtended  by  ds,  we  have 

cos  id  =  sin  i(p. 
Whence,  since  <p  is  very  small, 

Fig.  C  I\r=:2Ts'ini<P=   T4>=   T^.       (692) 

The  friction  due  to  iVis  then 


/N=fT 


and  this  being  the  increment  of  the  tension,  we  have 

,ds 


dT=fT 


R 


{^93) 


(694) 


FRICTION.  243 


Whence 


Integrating,  we  have 


or 


-^  =/^- (695) 


logT'^^'  +  logC, (696) 


T  —  Ce^ (697) 


When  J  =  o  we  have  T—  T^,  and  when  5  =  5  we  have  T  =.  T^\ 
therefore 

T,=  T,eR=.T,ef-n-, (698) 


n  being  the  number  of  times  the  cord  is  wrapped  around  the 
cylinder.     This  relation  may  be  written 

^^e^-"- (699) 

From  which  it  is  seen  that  as  the  number  of  turns  increases  in 
arithmetical  progression  7",  increases  in  geometrical  progression. 
We  see,  also,  how  it  is  possible  for  a  man  exerting  a  tension  T^ 
on  the  free  end  of  a  rope  wound  several  times  around  a  pile  to 
hold  in  equilibrium  the  very  much  greater  tension  7",  of  the 
other  end  caused  by  the  stoppage  of  a  boat  at  a  wharf. 


244  THEORY  OF  MACHINES. 


Stiffness  of  Cordage. 

197.  In  theoretical  mechanics  a  cord  is  defined  to  be  a  collec- 
tion of  molecules  so  united  as  to  form  a  perfectly  flexible  mate- 
rial line.  Considering  it  also  to  be  without  weight  and  to  be 
inextensible,  the  effect  of  a  force  is  supposed  to  be  transmitted 
along  its  length  without  loss. 

The  tension  of  a  cord  is  the  intensity  of  the  force  which  tends 
to  separate  any  two  of  its  adjacent  sections. 

Cordage  is  a  term  applied  to  all  varieties  of  lines,  cord,  and 
rope,  formed  by  twisting  together  the  textile  fibres  of  hemp,  flax, 
cotton,  etc.  Since  these  fibres  are  neither  perfectly  flexible  nor 
inextensible  themselves,  cordage  must  be  much  less  so,  and  hence 
will  offer  a  resistance  to  being  bent  from  the  direction  which  it 
naturally  assumes.  By  stiffness  of  cordage  is  meant  the  resistance 
which  it  offers  when  it  is  forced  to  take  a  curved  form  in  adapt- 
ing itself  to  the  surfaces  of  wheels  and  pulleys. 

The  law  of  this  resistance  has  been  deduced  by  Coulomb  from 
numerous  experiments  made  on  different  kinds  of  cordage.  He 
found  that  the  stiffness  of  cordage  is  composed  of  two  parts,  viz., 
one,  a  constant  depending  on  the  natural  torsion  of  its  fibres;  the 
other,  a  variable  depending  on  the  intensity  of  the  stretching 
force  applied  to  the  cord.  He  also  found  that  for  the  same  cord 
it  varies  inversely  with  the  diameter  of  the  wheel  around  which 
the  cord  is  bent.  If  ^S  be  the  stiffness,  K  the  constant  part  due 
to  the  natural  torsion  of  the  fibres,  /  that  due  to  a  unit  of  ten- 
sion, W  the  total  tension,  and  D  the  diameter  of  the  wheel, 
Coulomb's  experimental  law  for  a  particular  cord  is  expressed 
by  the  formula 

*5=— ^ (700) 

The  quantities  K  and  /,  according  to  Morin,  ought  properly 
to  be  expressed  in  terms  of  the  number  n  of  yarns  of  whicli  the 


STIFFNESS  OF  CORDAGE.  245 

rope  is  composed.  Making  use  of  Coulomb's  results,  Morin  found 
that  if  /  be  assumed  to  vary  with  «,  and  if  K  be  taken  to  consist 
of  two  terms,  one  proportional  to  n  and  the  other  to  «",  the  values 
of  6"  derived  from 


11 
»$■= -^(0.002148  +  O.OOI772W -|- 0.0026256  ^)     .     (701) 


would  conform  to  all  of  Coulomb's  results  for  ordinary  new 
white  rope,  and 

6*  = -^(0.01054  +  o.oo25«  +  0.003024  fF)   .     .     .     (702) 

to  those  for  tarred  rope.    These  formulas  of  Morin  are  identically 
those  of  Coulomb,  when  for  white  rope  we  place 

K  =  «(o.oo2i48  -|-  o.ooi772«)     and     /  =:  «(o.oo3024), 

and  for  tarred  rope 

J^  =  ^(0.01054  -|-  0.0025//)  ^"d     -^  —  «(o.oo3024). 

The  values  of  S  in  pounds,  for  both  kinds  of  rope,  bent  over  a 
wheel  or  axle  one  foot  in  diameter,  under  a  tension  of  one  pound, 
are  given  in  Table  VI.  For  other  axles  and  tensions  these  values 
substituted  in  the  above  formulas  will  give  the  desired  results. 

The  stiffness  of  partly  worn  or  oily  rope  is  less  than  that  of 
new  rope;  it  may  be  only  one  half  as  great.  The  *'  natural  stiff- 
ness" ^of  wet  rope  is  twice  that  of  dry,  while  the  value  of  /  is 
the  same  for  both. 

The  stiffness  of  Cordage  consumes  work  when  the  cord  is 
wound  on  the  wheel  so  as  to  adapt  itself  to  the  circumference. 
This  work  is  equal  to  the  product  of  the  intensity  of  the  resist- 
ance by  the  path  described  by  its  point  of  application,  estimated 


246  THEORY  OF  MACHINES. 

in  the  direction  of  the  resistance.  The  path  is  evidently  equal  ta 
the  length  of  that  portion  of  the  cord  wound  on  the  wheel,  which 
is  the  actual  distance  passed  over  by  the  resistance.  Then  if  it 
be  the  number  of  turns  of  the  wheel,  the  cord  wound  is  in  length 
equal  to  tmtR^  and  the  quantity  of  work  consumed  by  the  stiff- 
ness of  the  cordage  will  be,  for  new  white  rope, 

=  {K^  IW)n7t (7o3> 

From  this  we  see  that  the  work  consumed  in  n  revolutions 
is  independent  of  the  radius  of  the  wheel;  as  it  should  be,  since 
the  increased  stiffness  for  a  wheel  of  smaller  radius  is  compen- 
sated by  the  less  path  over  which  the  resistance  works  in  making 
one  complete  turn. 

Machines. 

198.  A  machine  is  any  instrument  or  device  designed  to 
receive  energy  from  some  source,  and  to  overcome  certain  resist- 
ances in  transferring  this  energy  to  other  bodies.  Every 
machine  consists  of  three  essential  parts,  viz.,  the  driving  point, 
the  working  point,  and  the  train.  The  first  is  the  point  at  which 
the  energy  is  received,  the  second  that  at  which  the  transmitted 
energy  is  applied,  and  the  third  is  the  series  of  parts  connecting 
the  first  and  second. 

Tlie  operating  forces  in  a  machine  are  classified  as  Powers 
and  Resistances.  A  power  is  a  force  which  increases  or  tends  to 
increase  the  momentum  of  the  parts  of  a  machine;  a  resistarice 
is  a  force  which  diminishes  or  tends  to  diminish  their  momentum. 
Those  resistances  which  the  machine  is  primarily  designed  to 
overcome  are  Ccilled  useful,  and  the  energy  expended  in  overcom- 
ing them  is  called  useful  work :  all  other  resistances  are  called 
prejudicial  or  wasteful,  and   the  energy  expended  in  overcoming 


MACHINES.  247 


them  is  called  wasteful  or  lost  work.  From  these  definitions  we 
see  that  the  action-lines  of  the  powers  must  either  coincide,  or 
make  acute  angles,  with  their  corresponding  virtual  velocities,, 
and  consequently  (Art.  68)  their  elementary  quantities  of  work 
will  be  positive;  and  that  the  action-lines  of  the  resistances  must 
either  be  opposite  to,  or  make  obtuse  angles  with,  their  respec- 
tive virtual  velocities,  and  hence  their  elementary  quantities  of 
work  will  be  negative.  Recalling  the  fundamental  principle  that 
enersry  can  neither  be  created  nor  destroyed,  we  see  that  in  the 
discussion  of  a  machine  it  is  necessary  to  ascertain  what  amount 
of  energy  it  has  received  from  the  source,  and  how  much  of  this 
has  been  exchanged  for  useful  and  lost  work,  and  how  much  still 
continues  in  tlie  machine  as  potential  or  kinetic  energy.  The 
energy  received  by  the  machine  is  generally  designated  as  the 
work  of  the  powers;  that  which  has  been  distributed  by  the  ma- 
chine to  masses  forming  no  part  of  the  machine  is  the  work  of  the 
resistances, 

199.   Theory  of  Machines. — Resuming  the  Equation  of  Energy, 

we  may  apply  it  to  any  machine  by  letting  P  and  dp  represent 

the  type-symbols  of  the  intensities  and  projected  virtual  veloci- 
ties of  the  powers,  Q  and  dq  those  of  the  resistances,  and  m  the 
mass  of  any  particle  of  the  machine;  then  we  have 

:2Pdp  -  :2Qdq  =^  ^m-^Js (704) 

Integrating,  we  have 

:2jpdp-:2jQdq^\^mv'^C    .     .     .     (705) 

for  the  general  equation  of  energy  applied  to  machines.  If  z\  be 
the  type-symbol  of  the  velocities  of  the  masses  j?i  at  the  instant 


248  THEORY  OF  MACHINES. 

tlie  machine  begins  to  receive  energy  from  the  source,  we  have, 
since  the  work  of  the  powers  and  resistances  is  then  zero, 

C  —  —  \^mv^ (706) 

Substituting  this  value  of  C  in  the  general  equation,  we  have 

^fPdp  -  ^jQdq  =  i2mv'  -  i:^mv;.  .     .      (707) 

If  the  machine  start  from  rest,  thenz;„=  o,  and  Eq.  (707)  becomes 

2rFdp-2fQdq  =  i:2mv' (708) 

If  the  integration  be  taken  between  any  limits  corresponding  to 
the  states  i  and  2,  Eq.  (705)  will  become 

^  rPdp  -  2  rqdq  =  \^mv^  -  i2mv,\      .     (709) 

The  theory  of  machines  is  embodied  in  the  general  equation 
(707),  and  may  be  derived  from  either  of  its  special  forms,  Eq. 
(708)  or  (709).  Taking  (709),  which  applies  to  any  machine  in 
operation,  we  see  that  if  its  kinetic  energy  increase  between  any 
two  successive  states,  the  increment  is  exactly  equal  to  the  ex- 
cess of  the  work  of  the  powers  over  that  of  the  resistances  in  the 
intervening  interval  of  time.  If  its  kinetic  energy  diminish,  then 
the  loss  is  equal  to  the  excess  of  the  work  of  the  resistances  over 
that  of  the  powers,  and  should  this  condition  continue,  the  ma- 
chine will  come  to  rest  when  the  total  kinetic  energy  is  wholly 
absorbed  in  making  good  the  deficiency.  If  there  be  no  change 
in  the  kinetic  energy  the  total  work  of  the  powers  is  exactly 
equal  to  that  of  the  resistances  during  the  interval  in  which  the 
kinetic  energy  is  invariable. 


MACHINES.  249 


Hence,  it  appears  that  when  the  work  of  the  powers,  in  any 
interval  whatever,  exceeds  that  of  the  resistances,  the  excess  is 
stored  up  as  kinetic  energy  in  the  masses  which  constitute  the 
machine;  and  when  more  work  is  required  by  the  resistances 
than  is  supplied  by  tlie  powers  in  a  given  interval  of  time,  the 
deficiency  is  made  good  by  the  withdrawal  of  kinetic  energy 
from  the  parts  of  the  maciiine.  The  total  quantity  of  work  done 
by  tlie  powers  from  the  instant  the  machine  starts  from  rest  un- 
til it  comes  to  rest  again,  or  from  any  particular  state  of  motion 
to  the  same  state  again,  is  precisely  equal  to  the  work  of  the  re- 
sistances in  this  interval.  Therefore,  whatever  energy  is  received 
by  the  machine  is  employed  in  making  resistances  perform  work, 
and  the  object  of  any  machine  is  to  get  as  much  useful  work  done 
by  the  expenditure  of  a  definite  quantity  of  energy  as  is  possil)!e. 
Whatever  kinetic  energy  remains  in  the  machine  at  any  instant 
is  simply  the  work  of  the  powers  which  has  not  heretofore  been 
used  in  overcoming  recurring  resistances,  and  hence  is  continu- 
ally accumulating,  to  be  afterwards  utilized  as  necessity  requires. 

200.  Use  of  Fly-7vheel. — Due  to  the  construction  and  applica- 
tion of  many  machines,  it  often  happens  that  the  energy  received 
from  the  source,  and  that  consumed  by  the  resistances,  vary 
with  the  time;  and,  in  addition,  these  variations  of  supply  and 
demand  may  neither  be  equal  nor  simultaneous.  In  such  cases, 
if  the  machine  be  of  relatively  small  mass  the  acceleration  of  ve- 
locity of  its  parts  will  be  correspondingly  great,  and  the  whole 
machine  will  be  subject  to  rapid  changes  of  motion,  wliich  are 
often  detrimental.  To  obviate  such  a  defect,  the  mass  of  the  ma- 
chine may  be  increased  by  the  addition  of  2i  fly-wheel.  This  con- 
sists of  a  mass  of  matter  distributed  in  the  form  of  a  ring  and 
suitably  connected  with  the  rotating  shaft  on  which  it  is  mounted. 
We  have  seen  that  the  kinetic  energy  of  rotation  is  measured  by 
^Qo^'^mr'^,  in  which  co  is  the  angular  velocity  and  "^Jtir^  the  mo- 
ment of  inertia  of  the  rotating  mass  with  respect  to  the  axis  of 
rotation.  Hence  the  changes  in  g?,  due  to  any  change  in  kinetic 
energy,  may  be  made  as  small  as  we  please  by  suitably  increas- 
ing ^Av/-";  this  may  be  done  by  increasing  either  the  mass  or  the 


250  THEORY  OF  MACHINES, 

radius  of  gyration  with  respect  to  the  axis.  By  the  introduction 
of  a  suitable  fly-wheel  the  changes  of  velocity  may  thus  be  di- 
minished to  any  desired  degree.  The  greater  the  moment  of  in- 
ertia of  the  fly-wheel,  the  greater  will  be  the  quantity  of  work 
which  it  will  store  up  for  a  given  increase  in  its  angular  velocity,, 
and,  similarly,  the  more  it  will  yield  for  a  given  decrease.  As 
the  change  of  level  in  a  reservoir,  due  to  the  addition  or  dis- 
charge of  a  given  quantity  of  water,  will  be  less  noticeable  as 
the  surface  area  is  greater,  so  likewise  will  be  the  changes  of 
velocity  in  the  moving  parts  of  a  machine,  due  to  a  difference 
between  the  work  of  the  powers  and  that  of  the  resistances,  ac- 
cording as  the  moment  of  inertia  of  its  fly-wheel  is  greater. 
From  this  analogy,  the  fly-wheel  may  be  regarded  as  a  reservoir 
of  work  in  a  machine. 

201.  Efficiency. — If  W  be  the  whole  amount  of  energy  sup- 
plied to  a  machine  in  a  given  time,  during  which  its  kinetic 
energy  remains  constant,  and  Wu  and  Wi  be  that  employed  in 
overcoming  the  useful  and  wasteful  resistances  respectively,  in 
the  same  time,  then 


W-Wu^  Wi. (710). 


The  ratio  of  the  total  work  to  that  of  the  useful  resistances,. 

or-^j^,  is  called  the  modulus  or  efficiency  oi   the  machine.     This 

ratio  is  evidently  always  less  than  unity,  which  is  its  maximum, 
limit,  and  which  it  can  never  reach,  since  Wi  can  never  in  actual 
machines  become  zero. 

Wi  can  be  diminished  in  value: 

(i)  By  avoiding  all  unnecessary  friction. 

(2)  By  diminishing  the  intensity  of  the  necessary  iv\Q.x!\ovi\  this. 
may  be  accomplished  by  selecting  material  for  the  contact  sur- 
faces whose  friction-coefficients  are  small,  or  reducirig  these  co- 
efficients by  the  application  of  lubricants. 

(3)  By  decreasing   the  moments  of  friction   in   rotating  parts,. 


SIMPLE  MACHINES.  2$! 

either  by  decreasing  the  coefficients  as  above,  or  by  shortening^ 
the  lever-arm  of  friction,  or  both. 

(4)  By  such  an  assemblage  of  parts,  arrangement  of  supports 
and  solidity  of  foundation,  as  to  avoid  sudden  and  unnecessary 
vibrations  and  shocks.  These  consume  work  which  is  dissipated 
in  the  form  of  heat-energy. 

The  principal  sources  of  energy,  whether  from  fuel,  air  in  mo- 
tion, animal  or  water  power,  etc.,  have  received  their  supply 
either  directly  or  indirectly  from  the  sun,  which  is  constantly 
parting  with  a  portion  of  its  energy  in  the  form  of  heat.  In  es- 
timating energy  the  foot-pound  has  been  assumed  as  the  unit. 
This  unit  does  not  take  into  consideration  the  time  in  which  the 
quantity  of  work  is  expended,  and  since  the  element  of  time  is 
important  in  the  use  of  machines,  a  different  unit  from  the  foot- 
pound is  required  in  measuring  their  efficiency.  Such  a  unit  is. 
the //^rj-f'/^Tf^'^r,  which  corresponds  to  the  expenditure  of  550  foot- 
pounds of  work  in  a  second  of  time.  An  engine  of  ten  horse- 
power is  one  which  is  capable  of  doing  5500  foot-pounds  of  work 
in  a  second. 


Simple  Machines. 

202.  The  Sifnple Mac/itneSy  or,  as  they  are  sometimes  called,  the 
Mechanical  Powers,  are  the  Cord,  Lever,  Pulley,  Wheel  and  Axky 
Inclined  Plane,  Wedge,  and  Screw.  All  other  machines  are  formed 
of  combinations  of  these,  and  when  the  relations  existing  between 
the  powers  and  resistances  are  known  in  the  simple  machines,  the 
corresponding  relations  in  compound  machines  may  be  derived. 
There  is  said  to  be  Si  gain  of  power  in  a  machine  when  the  inten- 
sity of  the  power  is  less  than  that  of  the  corresponding  useful 
resistance  with  which  it  is  compared;  and  a  loss  of  pouter  when  the 
intensity  of  the  power  is  greater  than  that  of  the  resistance. 
These  terms  are  technical,  and  are  used  merely  to  compare  the 
intensities  of  the  powers  and  resistances,  and  not  to  compare  the 
work  done  by  these  forces.  If  we  take  the  quantity  of  work  done 
by  the  power  and  by  the  useful  resistance  to  be  equal,  as  in  the 


252  THEORY  OF  MACHINES. 

limiting  case,  it  is  evident  that  when  there  is  a  gain  of  power  the 
path  described  by  its  point  of  application,  estimated  along  its 
action-line,  must  be  greater  than  that  of  the  resistance;  that  is, 
its  velocity  will  be  greater  than  that  of  the  point  of  application 
of  the  resistance,  since  these  unequal  distances  are  described  in 
the  same  time;  this  is  technically  called  a  loss  of  speed.  When 
there  is  a  loss  of  power  there  will  be  a  corresponding  gain  of 
speed. 

The  object  of  discussing  Simple  Machines  is  to  find  the  rela- 
tion existing  between  the  intensity  of  the  power  and  that  of  the  use- 
ful resistance.  To  do  this,  we  first  place  the  work  of  the  powers 
equal  to  the  sum  of  the  work  done  by  all  the  resistances;  the 
equation  so  formed  we  know  to  be  true,  whilst  the  parts  of  the 
machine  have  uniform  motion,  for  during  this  time  the  energy 
received  by  the  machine  is  wholly  absorbed  by  the  work  of  the 
resistances. 

This  condition  will  always  be  presupposed,  and  Eq.  (709)  will 
then  take  the  form 

2£Fdp  =  2£Qdg (711) 

We  are  then  often  able  to  eliminate  the  path  factor,  and  get  an 
equation  from  which  the  desired  relation  between  the  intensities 
may  be  obtained.  Passing  then  to  the  theoretically  perfect  ma- 
chine by  supposing  all  the  wasteful  resistances  to  be  neglected, 
we  find  the  theoretical  ratio  of  the  intensities  of  the  power  and 
useful  resistance. 

In  the  following  discussion  dp  and  dq  dive  taken  to  be  the  pro- 
jected elementary  paths  of  the  points  of  application  of  P  and  Q 
on  the  action-lines  of  these  forces  respectively,  in  the  time  dty 
during  which  the  forces  are  supposed  to  remain  constant  in  in- 
tensity; and  ds  is  the  path  described  by  a  point  at  a  unit's  dis- 
tance from  the  axis  of  rotation  in  the  same  time. 


THE  LEVER. 


253 


The  Lever. 
203.  The  lever  is  any  solid  bar,  straight  or  curved,  capable  of 
rotating  about  a  fixed  point  or  line  under 
the  action  of  a  power.  The  point  or  axis 
of  rotation  is  called  the  fulcrum.  Let  AB^ 
Fig.  70,  be  the  axis  of  a  lever,  O  the  axis  of 
the  trunnions  supporting  it,  P  and  Q  the 
power  and  resistance,  /  and  q  their  lever- 
arms  with  respect  to  (9,  and  r  the  radius  of 
the  trunnion.  The  resistances,  omitting 
that  of  the  air,  are  the  useful  resistance 
Q^  and  the  wasteful  resistance  friction  on 
trunnions,  whose  intensity  will  be  designated  by  F.  To  find  F, 
let  6  be  the  angle  included  between  the  action-lines  of  P  and  Q\ 
then  if  N  be  the  intensity  of  the  resultant  pressure  on  the  trun- 
nions, we  have 


Fig.  70. 


N1/P'-irQ^-\-2PQcoi  d (712) 


and 


F=N- 


f 


Vi+f 


(713) 


The  action-line  of  N  passes  through  C,  the  intersection  of  the 
action-lines  of  P  and  Q,  and  through  the  point  of  contact  of  the 
trunnion  with  the  fixed  support.  Hence  CN  is  the  action-line 
of  the  resultant  pressure.  The  lever-arm  of  friction  on  trun- 
nions is  r\  therefore  the  elementary  work  consumed  by  friction  is 


Frds', 


(7i4> 


that  absorbed  by  Q  is 


Q<lq=Qqds\ (715) 


254  THEORY  OF  MACHINES. 

^nd  that  done  by  the  power  P  is 

Fdp^^Ppds (716) 

Placing  the  elementary  work  done  by  the  power  equal  to  the 
sum  of  the  works  consumed  by  the  resistances,  under  the  sup- 
position that  the  motion  of  the  lever  is  uniform,  we  have,  after 
omitting  tlie  common  factor  ds^ 

Pj>  =  Qq^Fr,       , (717) 

Whence 

M+f  ? <'-« 


p 

Therefore  the  ratio  -^  of  the  lever  can  be  found  when  the  quan- 
tities P^  q,p,  r, /and  6  are  known. 

J? 
In  practice  both  factors  of  the  last  term  of  Eq.  (718),  —  and 

— ,  are  much  less  than  unity,  and  in  ordinary  cases  their  product 
/ 

is  negligible.     The  limit  of  the  ratio  of  the  power  to  the  resist- 
ance is  the  reciprocal  of  the  ratio  of  their  lever-arms;  or 

f =? ■  •  <'■'> 

Either  P  or  Q  may  be  taken  as  the  power  or  resistance,  but, 
to  agree  with  the  convention  established  heretofore,  the  power  is 
taken  to  be  that  force  whose  virtual  moment  is  positive.  When 
p>  q  there  is  a  gain  of  power,  and  when/  <  q  there  is  a  loss  of 
power. 

If  the  power  or  resistance,  or  both,  be  applied  in  a  plane  ob- 
lique to  the  axis  of  the  trunnion,  the  forces  must  be  resolved 


THE  LEVER. 


255 


into  components  parallel  and  perpendicular  to  the  axis.  The 
perpendicular  components  will  replace  P  and  Q  in  the  above 
discussion,  and  those  which  are  parallel  to  the  axis  will  either 
cause  motion  in  the  direction  of  the  axis  or  produce  pressure 
on  the  side  supports,  giving  rise  to  sliding  friction,  which  can 
readily  be  computed.  The  work  consumed  by  this  friction  will 
•appear  among  those  of  the  resistances  in  the  equation  of  equi- 
librium. 

Levers  are  commonly  divided  into  three  classes  or  orders. 
In  those  of  the  first  class  the  fulcrum  is  between  the  power  and 
resistance;  in  the  second  the  resistance  acts  between  the  fulcrum 
and  the  power;  and  in  the  third  the  power  is  applied  between 
the  fulcrum  and  the  resistance.  As  there  is  no  difference  in 
principle  in  these  orders  this  classification  is  unimportant. 

204.  The  principles  of  the  lever  are  involved  in  the  construc- 
tion of  the  common  balance.  To  find 
the  conditions  of  equilibrium,  let  O,  Fig. 
71,  be  the  point  of  suspension,  G  the 
•centre  of  gravity  of  the  balance  un- 
loaded, AC—CB—a,  OC  =  c,  OG=h. 
Let  the  balance  be  loaded  with  unequal 
-weights  Qy-  F,  and  suppose  that  it  has  q" 
taken  its  position  of  equilibrium  as  in 
the  figure.  Then  the  moments  of  the  forces  about  O  must  be 
•equal;  whence  we  have 


Fig.  71. 


Q{a  cos  ^  —  ^:  sin  ^)  =  F{a  cos  ^  +  ^  sin  6)  -{-wh  sin  ^, 


and  therefore 


tan  6  = 


c(Q -\- F)-h  7v/i' 


(720) 


The  conditions  required   in   a  balance  are  (i)  horizontality  of 
the  beam  when  the  arms  are  equal  in  length  and  the  weights  In 


256  THEORY  OF  MACHINES. 

the  scale-pans  are  equal;  (2)  sensibility,  which  is  estimated  by  the 
value  of  S  for  a  given  difference  in  the  weights;  (3)  stability ^  or 
the  tendency  to  return  to  horizontality  when  the  weights  are 
removed.  The  balance  is  so  constructed  that  the  first  condition 
is  satisfied  when  Q=^  P.  The  second  depends  on  the  greater  or 
less  value  of  6  for  a  small  value  oi  Q  —  P.  Assuming  a  constant 
difference  Q  —  F,  we  see  that  tan  6  will  increase,  and  hence  0 
also, 

(i)  when  a  is  large;  that  is,  when  the  arms  are  long; 

(2)  when  c  is  small,  that  is,  by  moving  the  point  of  suspension 
nearer  the  beam; 

(3)  when  P  -{-  Q\'i  small,  or  the  sum  of  the  weights  small; 

(4)  when  7v,  the  weight  of  the  balance,  is  small; 

(5)  when  h  is  small;  that  is,  when  G  is  not  far  below  the  beam. 
The  sensibility  of   the   balance   may  be  very  great  and   the 

balance  have  no  stability;  that  is,  no  tendency  to  return  to  its 
primitive  position  after  the  removal  of  the  weights,  and  the 
former  will  have  to  be  modified  to  satisfy  the  latter  condition, 
which  is  of  course  essential. 

The  stability  increases  with  OG,  and  the  sensibility  decreases 
as  OG  increases.  In  any  particular  case  the  conditions  of  stabil- 
ity and  sensibility  must  be  determined  by  the  uses  for  which 
the  balance  is  designed.  Thus  for  rapid  weighing  of  large 
masses,  where  great  accuracy  is  not  important,  the  stability 
must  be  great;  while  for  the  weighing  of  the  precious  metals, 
drugs  in  small  quantities,  etc.,  great  sensibility  is  of  primary 
importance.  There  is  generally  some  device  attached  to  the; 
balance  to  check  oscillations  when  the  stability  is  slight. 


THE  WHEEL  AND  AXLE.  257 


The  Wheel  and  Axle. 

205.  This  machine  consists  of  a  wheel  W^  Fig.  72,  firmly 
attached  to  a  cylinder  C,  whose  trunnion-ends  t  and  f  rest  on 
trunnion-b^s.  The  power /^  maybe  applied 
through  the  intervention  of  a  rope  passing 
over  a  groove  in  the  wheel,  or  by  a  crank,  or  "^TTTTTTTr 
by  capstan-bars;  its  action-line  is  generally 
tangent  to  the  circumference  of   the  wheel. 


in* 


4-10 


w 


m 


The  useful  resistance   Q  is   applied    tangen-  n* 
tially  to  the  cylinder  C  by  means  of  a  rope  £]q       L 

wound  upon  the  cylinder.     The  principal  re- 
sistances are  Q^  and  the  two  wasteful  resist-  ^'^  72. 
ances,  the  stiffness  of  cordage  caused  by  Q^  and  the  friction  on  trun- 
nions.    Let  R,  r  and  p  be  the  radii  of  the  wheel,  the  cylinder 
and  the  trunnions,  respectively;  then  the  elementary  work  of  the 
power  is 

Pdp^PRds\ (721) 

that  of  Q  is 

Qdq  =  Qrds\ (722) 

that  of  stiffness  of  cordage  is 

^£±I^rds^\l,K^IQ)ds,   .     .     .     .     (723). 

the  proper  values  of  jFT  and  /  being  assumed  for  the  kind  of  rope 
used;  that  of  friction  on  trunnions  is 

f{Ar^jv^)pds, (724) 

in  which  N"  and  iV'  are  the  pressures  at  /  and  t*  due   to  w,  the 
17 


258  THEORY  OF  MACHINES. 

weight  of  the  machine,  and  to  the  forces  P  and  Q\  and/'  is  a 
symbol  for  -  Placing  the  work  of  the  power  equal  to 

the  sum  of  the  works  of  the  resistances,  and  omitting  the  com- 
mon factor  ds,  we  have 


PR^Qr^\{K-^IQ)^f{N^N')fi.    .     .     (725) 

The  limiting  ratio  of  the  power  to  the  useful  resistance,  ob- 
tained by  neglecting  friction  on  trunnions  and  stiffness  of  cord- 
-age,  is 

P        r  .     .. 

Q=R'-    ■    ■ (7^') 

the  same  as  in  the  lever;  as  it  should  be,  since  the  principle  of 
the  two  machines  is  essentially  the  same. 

To  find  the  resultant  pressures  iV^and  iV'  at  /  and  /',  let  /be 
the  length  of  the  axis  between  the  middle  points  of  the  trunnions, 
at  which  iVand  iV'  may  be  supposed  applied;  let  a,  b  and  c  be 
the  distances,  estimated  parallel  to  the  axis,  from  the  point  of 
•application  of  iV  to  the  action-lines  of  Q^  w  and  P,  respectively; 
the  corresponding  distances  of  N'  will  be  I  —  a,  I—  b  and  l  —  c. 
Let  the  action-lines  of  w  and  Q  be  vertical,  and  let  that  of  P 
make  an  angle  0  with  the  vertical.  The  components  of  P  will 
then  be  P  cos  0,  vertical,  and  P  sin  0,  horizontal.  By  the  prin- 
ciples of  parallel  forces,  the  components  of  the  vertical  forces  g, 
m  and  P  cos  0,  at  /,  will  be 

Q — - — ,     w — —-     and    P — j—  cos  0, 

and  the  component  of  P  sin  <p  at  t  will  be 

P—j —  sm  0; 


THE  WHEEL  AND  AXLE.  259 

and  hence  the  resultant  N  will  be 

N  =^-j  '^\Q(l-a)^w{l-b)^P(l-c)  cos  0]'+/''(/-^)»sin'0.(727) 
Similarly  the  vertical  components  of  N'  at  f  will  be 


Q—^     w—     and     P—  cos  0; 
I  I'  I 


and  the  horizontal  component  will  be 

P-j  sin  0. 
Whence 

iV^'  =  L  ^(Qa  ^wb^  Pc  cos  0)'  +  P\''  sin"  0.       (728) 
If  P  be  vertical,  0  =  o,  and  we  have 

7\^=i[e(/-a)  +  ^(/-^)  +  ^(/-.)];   ] 

}•  •    (729) 
N'  =  -{Qa^wb  +  Pc);  \ 

whence 

N^N*  ^w^Q-\-P (730) 


By  substituting  the  values  of  jV  and  W  in  Eq.  (725)  an  ap- 

p 
proximate  value  of  the  ratio  -^-  may  be  found. 


26o  THEORY  OF  MACHINES. 

206,  The  Differential  Wheel  and  Axle. — Eq.  (726)  shows  that 
the  gain  of  power  increases  as  r  diminishes;  but  since,  owing  to 
the  stress  of  Q,  r  cannot  be  made  very  small  with- 
out the  liability  of  bending  or  breaking  the  axle, 
there  is  a  practical  limit  to  the  gain  of  power  in  the 
ordinary  wheel  and  axle.  In  the  differeniial  wheel, 
Fig.  73,  the  axle  consists  of  two  cylinders  of  differ- 
ent radii.  The  resistance  is  attached  to  a  movable 
pulley,  and  a  continuous  cord  is  wound  oppositely 
on  the  two  cylinders  of  the  axle,  after  partially  en- 
veloping the  pulley.  Supposing  the  power  to  be 
Fig.  73-  applied  tangentially  to  a  wheel,  whose  radius  is  -/?, 
mounted  on  the  axle,  we  will  have  by  the  equality  of  moments, 
neglecting  the  wasteful  resistances,  and  calling  r  and  r'  the  radii 
of  the  cylinders, 

FR^\Q{r-r% (731) 

or 

P       r  —  r* 


Q  2R 


(732) 


Hence  if  r'  be  made  great  enough  to  withstand  any  possible 
stress  of  Q,  the  difference  r  —  r'  may  be  made  as  small  as  we 
please,  and  thus  give  any  desired  gain  of  power.  A  combination 
similar  in  principle  is  used  to  lift  very  heavy  projectiles  to  the 
muzzle  of  the  gun. 


The  Pulleys. 

207.  (i)  The  Fixed  Pulley. — This  consists  essentially  of  a 
grooved  wheel  supported  on  trunnions  about  which  it  can  turn 
freely,  friction  being  disregarded.  The  power  and  useful  resist- 
ance are  applied  at  the  ends  of  the  same  cord,  which  partly  en- 
velops the  wheel,  and  is  prevented  from  slipping  by  the  friction 


THE  PULLEYS.  26 1 


between  the  cord  and  wheel.  If  in  the  wheel  and  axle  the 
radius  of  the  cylinder  be  taken  the  same  as  that  of  the  wheel,  it 
becomes  essentially  a  fixed  pulley,  and  therefore  we  may  make 
the  equation  deduced  for  the  former  applicable  to  the  latter  by 
making  the  proper  changes  in  the  quantities  which  enter  it. 
Hence,  for  the  fixed  pulley,  Fig.  74,  since 


j^  =  r,     and     «  =  /^  =  ^  =  -J/, 
Eq.  (727)  becomes 


N  =  N'=  iV{w-\-Q-^F cos  0)^4-  F'  sin»  0,  (733)  pxc.  74. 

and  Eq.  (725), 

FR=QRJ^i(^K^IQ)+f2Np',    .     .     .     (734) 


whence 


^-^  +  ^i^+/'l^^-   •  •  •  •  (^35) 


Neglecting  stiffness  of  cordage  and  friction  on  trunnions,  the 
limiting  ratio  of  the  power  to  the  useful  resistance  becomes, 
Eq.  (735)» 

^=^ (736) 


Theoretically,  then,  the  least  value  of  the  power  is  equal  to  the 
useful  resistance,  and  the  fixed  pulley  is  used  simply  to  change 
the  direction  of  the  action-line  of  the  power. 

The  factor/'--  of  the  last  term,  Eq.  (735),  is  generally  very 


% 


262  THEORY  OF  MACHINES. 

small,  and  a  sufficiently  approximate  value  for  this  term  may 
be  obtained  by  putting  Q  for  F  in  the  value  of  N,  and  omitting 
«',  the  weight  of  the  pulley.  This  being  done,  we  have,  Eq.  (733), 

N=\Q  4/2(1  +^^)  =  Q  cos  i0.      .     .     (737) 

Let  Q  be  the  arc  of  the  wheel  enveloped  by  the  cord;  then 

cos  -^-0  =  sin  -J^, 
and 

N^Q^m^e .     (738) 

Substituting  this  in  Eq.  (735),  we  have 

^=C(i  +  2/'|sinie)  +  :^±^    .     .     .     (739) 

for  the  nearly  exact  relation  between  the  power  and  useful  re- 
sistance in  the  fixed  pulley. 

If  P  and  Q  be  parallel,  6  =  180°,  and  Eq.  (739)  becomes 


which,  since  the  coefficient  of  Q  and  the  second  term  are  con- 
stant for  the  same  rope  and  pulley,  may  be  written 

P  =  cc\fiQ (741) 


THE   PULLEYS. 


208.  (2)  The  Movable  Pulley  (Fig.  75). —  t 
When  one  end  of  the  cord  is  attached  to  a 
fixed  point,  and  the  useful  resistance  ^  is  con- 
nected directly  with  the  pulley,  the  latter  is 
called  a  movable  pulley.  The  resistance  in  this 
case  is  equal  to  2N^  and  if  the  weight  w  of 
the  pulley  be  neglected,  we  have,  Eq.  (738), 


whence 


N  =  \W=:  (2  sin  J^; 


W 


2  sin  id' 
Substituting  this  value  for  Q  in  Eq.  (739),  we  have 


K-\-I- 


W 


2R 


^       ^\2smie^^  R]^ 
or,  since  6  is  usually  180°, 


(742) 


(743) 


for  the  relation  between  the  power  P  and  the  useful  resistance 
W,  in  the  movable  pulley. 

If  stiffness  of  cordage  and  friction  be  neglected,  the  limiting 
ratio,  obtained  from  Eq.  (742),  is 


W=:P2S\nid=:P~', 


(744) 


in  which  C  is  the  chord  of  the  arc  enveloped  by  the  rope.     There- 
fore, neglecting  wasteful  resistances,  in   the  movable  pulley  the 


264 


THEORY  OF  MACHINES. 


power  is  to  the  useful  resistance  as  the  radius  of  the  pulley  is  to  the 
chord  of  the  enveloped  arc.  When  the  arc  is  between  0°  and  60°, 
and  between  300°  and  360°,  there  is  a  loss  of  power;  and  when 
between  60°  and  300°,  there  is  again  of  power;  the  greatest  gain 
of  power  is  at  180°  and  the  greatest  loss  at  0°  or  360°. 

209,  (3)  The  Block  and  Fall. — A  set  of  two  blocks  of  pulleys, 
connected  by  a  continuous  cord,  arranged  to  pass 
alternately  from  a  pulley  of  one  block  to  one  of 
the  other  as  in  Fig.  76,  is  called  a  block  and  fall ^  or 
a  block  and  tackle.  One  of  the  blocks  is  attached  to 
a  fixed  point,  and  the  other  to  the  useful  resist- 
ance W.  To  find  the  relation  between  F  and  W^ 
let  /j,  Z^,  ^31  .  .  .  4,  and  F^  be  the  successive  tensions 
on  the  straight  portions  of  the  cord,  t^  being  that 
on  the  first  portion  of  the  cord  which  is  attached 
to  one  of  the  blocks;  let  R  be  the  radius  of  each 
pulley,  and  p  that  of  each  pulley  trunnion,  and 
take  \6  to  be  90°.  Then  the  relations  between  the 
successive  tensions  will  be  given  by  Eq.  (741),  in 
which  Q  will  be  in  succession  Z^,  /,,,  etc.,  and  F 
will  be  successively  Z^,  /g,  /^,  etc.,  and  a  and  ^  are 
easily  determined  constants  depending  on  the  rope 
Fig.  76.       and  pulleys.     Then  we  have 


/,  =  ^  +  §t^ 
t^  =  a-^ftt, 

tn=a^  ptn 
F  =  a-^fitn 


-^^  +  ^A; 


^ 


=  ai^-^^t. 


1-13 


I  —  6^  ~  ^ 


fi 


\  —  8" 
^a- A-  +  M; 


1-/S 


(745) 


and  also 


THE  PULLEYS.  265 


Whence  we  have 

Substituting  this  value  of  t^  in  the  expression  for  4,  Eqs.  (745), 
we  have 

which  substituted  in  the  last  of  Eqs.  (745)  gives 

^  = '»+ .  =  ^^^ V-^  +  Ha^  -  ;«:r-J-    (749) 

In  the  case  illustrated  in  the  figure  ;?  =  4,  and  hence 

If  Stiffness  of  cordage  and  friction  be   neglected,  then  a  =  o 
and  >5  =  I,  and  we  have 

7P  =  «^ (751) 

that  is,  the  limiting  ratio  of  the  power  to  the  useful  resistance  is 


266 


THEORY  OF  MACHINES. 


r^ 


equal  to  the  reciprocal  of  the  number  of  the  parallel  portions  of 

the  cord  which  support  the  resistance. 

210.  (4)  Other  Cotnbinations  of  Fixed 
I  and  Movable  Pulleys. — The  value  of  the 
limiting  ratio  of  the  power  to  the  use- 
ful resistance  depends  only  on  the  num- 
ber of  fuovable pulleys,  the  arrangement  of 
the  cord,  and  the  method  of  attaching  it  to 
the  resistance  ;  an  inspection  of  the  com- 
bination is  generally  sufficient  to  es- 
tablish the  required  relation.  Thus 
if  ;/,  Fig.  77,  be  the  number  of  pulleys 


r^ 


9 


L 


m 


A  in  the  first  combination,  this  ratio  is 
readily  seen  to  be 


Fig. 


and  in  the  second  combination, 


p 

I 

w 

2»  —   I 

p 

I 

w~ 

2«—   I 

{752) 


(753) 


The  Inclined  Plane. 


211.  Replacing  /  in  Eqs.  (673)  and  (672)  by  P*  and  P",  we 
have 

sin  / -|-y  cos  i 


P'  =  IV 
P''  =  W 


cos  6'-|-/sin  0' 
sin  /  —/cos  / 


cos  8  —fs'in  & 


(754) 
(755) 


Considering  the  inclined  plane  as  a  machine,  the  first  equation 
expresses  the  relation  of  the  power  P  to  the  resistance  of  a 
body's  weight  W,  when  the  body  is  either  in  uniform  motion  up 
the  plane  or  in  a  state  bordering  on  such  motion;  and  the  second 


THE  INCLINED  PLANE. 


267 


equation  gives  the  relation,  when  the  body  is  in  uniform  motion 
down  the  plane  or  in  a  state  bordering  on  such  motion.  The 
difference  of  these  intensities,  for  the  same  weight  and  angles,  is 


F'  -P''  =  W- 


2/ cos  (/+6/) 


cos 


e-f 


sin 


■  •  (756) 


hence,  if  the  body  be  in  a  state  bordering  on  motion  up  the 

plane,  P'  maybe  diminished  in  intensity 

by  this  value  before  the  body  reaches  the 

state  bordering  on  motion  downward. 

Considering  W^  Fig.  78,  as   the  only 

resistance,    the    limiting    value    of    the 

P 
ratio  -77^,  obtained  by  making  /  =  o  in 

either  equation,  is 


P_ 

W 


sm  / 
cos  Q' 


Fig.  78. 


(757) 


When  d  is  zero  the  power  acts  parallel  to  the  plane  and  up- 
ward, and  we  have 

P         .     .      h 


-^  =  sin.  =  -; 


(758) 


that  is,  the  power  is  to  the  resistance  as  the  height  of  the  plane 
is  to  its  length,  and  there  is  always  a  gain  of  power. 

When  i)  =  360°  —  /  the  power  acts  horizontally,  pressing  the 
body  against  the  plane,  and  we  have 


■w='^'"'=P 


(7S9) 


that  is,  the  power  is  to  the  resistance  as  the  height  of  the  plane 
is  to  its  base;  and  there  is  a  gain  of  power  when  the  plane  has  a 
less  inclination  than  45°,  and  a  loss  of  power  for  greater  inclina- 
tions. 


268 


THEORY  OF  MACHINES. 


212.  The  elementary  quantity  of  work  expended  by  the 
power  P  in  moving  the  body  uniformly  up  the  plane  is,  when 
the  action-line  of  F  is  parallel  to  the  plane,  Eq.  (yii), 


Pds  —  Wds  sin  i  -\-fWds  cos  / 
=  Wdh^fWdl;    .     .     . 


(760) 


in  which  d/  and  dh  are  the  horizontal  and  vertical  projections  of 
ds.     Integrating  between  any  limits,  we  have 


J  Pds  =  W(h  -  h')  -{-fW{l-  /'); 


(761) 


or,  the  total  quantity  of  work  is  equal  to  the  work  stored  as  po- 
tential energy  of  the  weight,  plus  the  work  consumed  by  friction 
due  to  the  weight  over  a  path  equal  to  the  horizontal  projection 
of  the  actual  path  of  the  body. 


^r^. 


The  Wedge. 

213.  The  wedge  usually  consists  of  a  solid  triangular  prism, 
as  ABC  (Fig.  79),  which  is  inserted  into  an  opening  between 
two  bodies  or  parts  of  the  same  body,  to 
split  or  separate  them.  The  surface  AB, 
to  which  the  pressure  or  blow  is  given,  is 
called  the  back;  the  surfaces  AC  and  BC, 
the  faces;  and  their  line  of  intersection  C, 
the  edge  of  the  wedge. 

Let  the  wedge  ABC  be  inserted  within 
the  jaws  of  the  opening,  and  be  in  contact 
with  them  along  lines  projected  in  ;;/  and 
;;/.  Suppose  the  normal  pressures  iV^  and 
N'  and  the  iovctP  to  lie  in  the  same  plane, 
and  the  latter  to  be  normal  to  the  back  of 
the  wedge.  If  the  wedge  move  forward,  or  be  in  a  state  border- 
ing on    motion  forward,  the  friction  between  the  jaws  of  the 


/ 

\ 

^«: 

Jfn^ 

\ 

r- 

\ 

^W 

'1 

W 

' 

Fig.  79. 


THE  WEDGE.  269 


opening  and  the  surface  of  the  wedge,  due  to  the  normal  pres- 
sures iV^and  N\  will  act  from  m  and  ///  towards^  and  B  respec- 
tively. If  the  wedge  fly  back,  or  be  in  a  state  bordering  on  mo- 
tion outward,  the  friction  will  oppose  this  motion  or  tendency, 
and  act  along  the  faces  toward  C,  Considering  the  first  case, 
and  supposing  that  the  wedge  is  in  equilibrium  due  to  the  forces 
acting,  we  have,  for  the  components  parallel  to  AB, 

iV/sin  e  -  N  QO^d-\-  N'  cos  S'  -  iV'/ sin  l9' =  o;     (762) 

and  for  those  parallel  to  DC,  perpendicular  to  AB^ 

F  ^  Ns'md  ^  Nf  cos  d  -  N'  sin  d'  -  N'f  cos  <9'  =  0.(763) 

Eliminating  iV^'  from  these  equations,  and  representing  the  angle 
of  the  wedge  by  g?  =  ^  -f  6^',  we  have 

/^(cos  6'  -fsin  d') 
(i  -/=')  sin  CD  +  2/ cos  ca'      •     •     •     ^'^ ^^ 

and  similarly  eliminating  N^  we  have 

N'=,  ^ir'~<:'"/^  ■  •  •  •  (765) 

(i  — /')  sin  G?+ 2/C0SG?  ^'   *'-' 

From  these  values  we  have 

_  W[(i  — /')  sin  GO  -\-  2/cos  GO  ] 
cos  6'  -/sin  d' 


_  N'\{i  — /')  sin  G0-\-  2f  cos  go] 
~~  cos  6  —  /  sin  6 


(766) 


In  order  that  P  may  have  a  possible  value  for  a  state  border- 
ing on  motion  forward  we  must  have 

cos  ^' >  /  sin  6^'     and     cos6'>/sin^, 


2/0  THEORY  OF  MACHINES. 

or 

cot  6' >  /    and     cot6/>/.     .     .    ,     .     (767) 

Substituting  for/  its  value  tan  a,  we  have 

cot  6^  >  tan  a     and     cot  6  >  tan  or, 
or 

^'  <  90°  -  a     and     6  <  go°  -  a.     .     .     .     (768) 

Hence 

CO  <  180°  —  2a (769) 

That  is,  z'n  order  that  the  wedge  may  be  driven  171,  the  angle  of  the 
wedge  must  be  less  than  180°  diminished  by  twice  the  a?igle  of  frictiofi. 

If  the  wedge  be  in  a  state  bordering  on  motion  outward,  the 
friction  terms  in  Eq.  (766)  will  change  their  signs,  and  we  have 

_  iV[(i  —  f)  sin  GO  —  2f  cos  gd\ 

~  cos  ^'+7  sin  d' 

__  iV^'[(i  —  f^)  sin  GO  —  2/  cos  gd\  .       . 

~  cos6>+/sin  6  •      •     •     ^^^°^ 

If  some  pressure  be  required  to  prevent  the  wedge  from  flying 
out,  we  must  have  , 


(i  —f^)  sin  00  >  2/ cos  GJ, 


or 


tan  a  .       . 

tanc»>2 r— ^— ; (771) 

I  —  tan   a  \i  >   1 

which  reduces  to 

tan  00  >  tan  2af, 

or 

03  >  2a (772) 


THE   SCREW.  271 


Hence  we  see  that  in  order  that  the  wedge  may  be  held  in  its 
place  when  the  external  pressure  is  removed^  the  angle  of  the  wedge 
7mcst  be  less  than  twice  the  angle  of  friction. 

When  the  wedge  is  used  as  a  power  iVand  N'  are  generally 
nearly  equal,  and  either  may  be  considered  as  the  resistance  to 
be  overcome.  The  particular  form  of  the  wedge  in  any  case  de- 
pends on  the  special  use  for  which  it  is  intended.  Thus  in  split- 
ting wood  it  is  usually  made  isosceles,  and  the  power  P  is  ap- 
plied to  the  back  of  the  wedge  as  an  impulsion.  The  axe,  chisel, 
engraver,  knife,  tool  of  a  plane,  and  the  raised  projections  of  a 
file,  are  examples  of  wedges,  whose  forms  are  modified  in  accord 
with  the  above  principles,  for  the  particular  purposes  for  which 
they  are  designed.  Taking  the  wedge  to  be  isosceles  and  N  — 
N\  we  have  for  the  ratio  of  P  Xo  N 


-P  _  (i  -/')  sin  GO-\r  2/coscj^ 

N  cosica-Zsin^G?        »      •     •     •     \11^) 


and  omitting  friction, 


=  2  sin  i(i? (774) 


N       cos  \0D 

Hence  the  gain  of  power  increases  very  rapidly  as  the  angle 
of  the  wedge  diminishes. 

The  Screw. 

214.  The  screw  combines  the  principles  of  the  lever  and 
inclined  plane.  It  consists  usually  of  a  solid  circular  cylinder, 
called  the  newel,  on  the  surface  of  which  is  a  thread  or  fillet, 
whose  section  by  a  plane  through  the  axis  of  the  cylinder  is 
usually  either  a  rectangle  or  a  triangle.  The  thread  of  the  screw 
is  a  volume  which  may  be  generated  by  a  rectangle  or  triangle 


272 


THEORY  OF  MACHINES. 


Fig.  8o. 


Iiavipig  its  base  on  the  cylindrical  surface  and  always  parallel  to 
the  axis  of  the  newel,  moving  uniformly 
around  and  along  the  axis.  Every  point 
of  the  generating  area  will  therefore 
describe  a  helix,  and  the  upper  and  under 
sides  will  describe  helicoidal  surfaces^ 
The  distance  between  the  successive  posi- 
tions of  the  same  point  of  the  generating 
area,  measured  in  the  direction  of  the 
axis  of  the  newel,  after  one  complete  rev- 
olution, is  called  the  pitch  of  the  screw, 
or  the  helical  interval.  In  screws  with  rect- 
angular threads  the  pitch  must  be  at  least 
equal  to  twice  the  base  of  the  generating 
rectangle;  in  triangular  threads  it  is  usu- 
ally equal  to  the  base  of  the  generating 
triangle. 

The  screw  is  engaged  in  a  nut  whose  interior  cylindrical  sur- 
face is  screw-cut  in  such  a  manner  as  to  fit  the  fillet  accurately. 
The  useful  resistance  to  be  overcome,  if  the  nut  be  fixed  in  posi- 
tion, is  applied  to  the  foot  of  the  screw  so  that  its  action-line 
may  be  in  the  direction  of  the  axis  of  the  newel  ;  if  the  nut 
have  freedom  of  motion  and  the  screw  is  fixed,  the  useful  resist- 
ance is  applied  to  the  nut. 

Take  the  axis  of  the  newel  as  the  axis  of  z^  and  let  abc^  Fig. 
80,  be  the  generating  area.     Let 

P,  be  the  constant  angle  made  by  ab  in  all  of  its  positions 
with  z. 

r,  the  distance  of  any  helix  from  the  axis,  constant  for  the 
same  helix,  but  variable  for  different  helices. 

y^  the  constant  angle  made  by  any  assumed  helix  with  the 
horizontal  plane. 

0,  the  angle  through  which  the  screw  or  nut  is  rotated. 

/,  the  lever  arm  of  the  power. 

For  the  elementary  work  of  the  power  we  have 


FdJ>  =  Fld(P, (775) 


THE   SCREW.  273 


and  for  the  work  of  the  useful  resistance,  Q, 

Qdz  =  Qrd^  tan  y  =  Qr^d<p  tan  7',  .     .     .     (776) 

in  which  r'  is  the  radius  of  a  mean  helix,  and  ;/'  the  angle  which 
this  helix  makes  with  the  horizontal. 

Let/ be  the  coefficient  of  friction,  and  iVthe  normal  pressure, 
and  suppose  the  friction  concentrated  on  the  mean  helix  whose 
radius  is  r\     Then  we  have  for  the  elementary  work  of  friction 


Hence 


or 


Pld4,=  Q,r'd<t,X^x.Y'^~Y'     ■     •     ■     ^"^^ 


215.  To  find  the  relation  between  jP  and  Q  it  is  necessary  to 
find  JV  in  terms  of  Q.  From  the  equilibrium  of  the  forces  we 
know  that  the  algebraic  sum  of  their  intensities  in  any  direction 
is  equal  to  zero.  Hence,  resolving  the  forces  P,  Qy  N  and  fN'^ 
we  have,  for  the  sum  of  their  components  in  the  direction  of  the 
axis  2, 

Q  -\-fJV sin  y'  —  NCOS  d^  =  o,  .     .     .     .     (780) 

• 
in  which  dz  is  the  angle  made  by  the  normal  to  the  helicoidal 
surface,  at  the  assumed  point  of  equilibrium  on  the  mean  helix, 
with  the  axis  of  z.     The  cosine  of  this  angle  is 


'"'^'^^irr^^yT^j'  ■  ■  •  ^'"^ 


iR 


274  THEORY  OF  MACHINES, 

hence  Eq.  (780)  becomes 

e  =  i\^(cos  ^^-/sin /),       ....     (782) 


or 


^=cosg,-^/sin/ (7«3) 

Substituting  this  value  of  iV^in  Eq.  (779),  we  have 

_  Qr'l^ny'  I / \ 

/  V    "^  sin;/' (cos  ^,-/ sin/)  j ^^^^ 

Replacing  cos  Bz  by  its  value,  and  reducing,  we  have 

p  ^  g^^'tan//  /4/i  +  tan>'  +  cot-^  \ 

/  \         sin/-/sinV|/i  +  tanV  +  cotVV 

If  the  thread  be  rectangular,  /?  =  90°,  and  we  have 

^^g^a^Y^  /^FWT^V     .     .     (,86) 

^  \  sin  ;/' — /sin^;/   r  I  +  tan'' ;/'/ 

hence,  all  other  things  being  equal,  the  screw  with  a  rectangular 
thread  is  more  advantageous  than  one  with  a  triangular  thread. 
If  we  suppose  the  friction  to  be  neglected,  then/=  o,  and  the 
limiting  ratio  of  the  power  to  the  resistance  is 


P  _r  tan  y 


(7S7) 


Q         I 

Multiplying  and  dividing  the  second  member  by  2K,  we  have 

P_^^anr       (^88) 

Q  27tl 


THE   CORD. 


275 


or  when  friction  is  neglected  the  power  is  to  the  useful  resistance  as  the 
helical  interval  is  to  the  circumference  described  by  the  extremity  of  the 
lever  arm  of  the  power;  hence  there  is  usually  a  great  gain  of 
power  in  the  application  of  this  machine. 


The  Cord. 

216.  Let  a  perfectly  flexible  and  inextensible  cord  assume  a 
position  of  equilibrium  under  the  action  of  any  forces  whatever. 
The  resultant  of  the  forces  acting  at  either  extremity  must  be  in 
the  direction  of  the  cord  at  that  extremity;  for,  if  it  have  a  com- 
ponent perpendicular  to  the  cord,  the  latter,  being  perfectly 
flexible,  must  move  in  the  direction  indicated  by  the  perpendic- 


FlG.  81. 

ular  component.  Let  i  2  3,  Fig.  81,  be  the  cord,  the  resultant 
R^  being  in  the  direction  2  i.  Since  the  point  of  application  of 
a  force  may  be  taken  to  be  any  point  of  its  action-line  within  the 
limits  of  the  body  on  which  it  acts,  the  resultant  R^  maybe  con- 
sidered as  applied  at  the  point  2.  Then  the  resultant  of  all  the 
forces  acting  at  2,  including  ^,,  must  be  in  the  direction  3  2;  and 
this  resultant,  i?„  may  be  considered  as  applied  at  the  point  3. 


2'J^  THEORY  OF  MACHINES. 

Thus  in  any  case  each  successive  resultant  may  have  its  point 
of  application  transferred  until  all  the  action-lines  have  a  com- 
mon point,  and  the  conditions  of  equilibrium  will  be  the  same 
as  before.  Therefore  the  conditions  of  equilibrium  for  a  perfectly 
flexible  and  inextensible  cord,  under  the  action  of  any  forces  what- 
ever, are  the  same  as  if  all  the  forces  were  applied  at  a  single  point, 
their  intensities  and  directions  remaining  unchanged. 

2I7.  Let  T^  be  the  tension  of  the  cord  at  the  origin,  assumed 
at  any  point;  /,  the  type-symbol  of  the  intensities  of  the  extrane- 
ous forces;  T,  the  tension  at  any  point;  and  let  d^,  By,  dz\  oc,  /?, 
y\  0j;,  0jj/»  0z>  be  the  angles  which  T^,  I  and  T  make  with  the 
co-ordinate  axes,  respectively.  Then  from  the  equilibrium  of  the 
svstem  we  have 


T  cos  0^  =  7;  cos  6^  +  ^7  cos  or;  \ 

Tcos  (f)y  =  7;  cos  6y  -{-  2/cos  fi;l    .     .     .     (789) 

Tcos  d)^  =  T  cos  e^  4-  ^7 cos  y;  ) 


the  last  terms  comprising  all  the  extraneous  forces  between  the 
origin  and  the  point  where  the  tension  is  T. 

If  forces  act  at  all  points  of  the  cord  in  such  a  manner  as  to 
make  the  tension  vary  by  continuity,  then  the  cord  will  assume 
the  form  of  a  curve,  and  Eqs.  (789)  become 


dx 
T—  =  7;  cos  6^-^21  cos  a; 

T^  =  7;  cos  dy  +  :5'7cos  /?; 


ds~  -^^  —  -y 

dz_ 

ds 


dz 
T—  =  r„  cos  e,  4-  :2'7cos  y. 


(790) 


Such  a  curve  is  called  a  funicular  curve^  and  Eqs.  (790)  are  its 
differential  equations. 

If  the   extraneous  forces   be  parallel  and  coplanar,  we  may 


THE   CORD. 


277 


assume  the  curve  in  the  plane  xz^  with  the  forces  parallel  to  z^ 
and  we  have 


hence 


cos  a  =  cos  a:"  =  etc.  =  o; 


dx  ^ 

T—z-  =  T^  cos  ^x  =  a  constant; 


(791) 


that  is,  the  componefit  of  the  tension  perpendicular  to  the  direction  of 
the  forces  and  in  their  plane  is  constant. 

218.  Let  the  cord  abed,  Fig.  82,  be  in  equilibrium,  the  length 
of  each  branch  representing  its  own  tension,  and  the  tensions  being  as- 
sumed constant  throughout.     Let  R^  and  R^  be  equal  and  opposite 


Fig.  82. 


to  the  resultants  of  the  forces  acting  at  the  points  b  and  c,  and 
let  the  symbols  (/,  /,),  (/,  R^,  etc.,  represent  the  angles  made  by 
the  corresponding  lines  in  the  figure.  When  three  forces  acting 
at  a  single  point  are  in  equilibrio,  their  intensities  are  inversely 
as  the  sines  of  the  opposite  angles,  and  we  have 


sin(/,^,)       sin(/,i?,)       sin  (tf^     ' 


.     (792) 


R. 


sin  (/',i?J       sin  (/,/?J       sin  (/,/,)'     ' 


.     (793) 


278  THEORY  OF  MACHINES. 

and  since  the  tensions  are  equal, 

sin  (t,R,)  =  sin  (/.^,);  )  ,      . 

sin  (t,J{,)  =  sin  (/,Ji,).  f ^'^'^^ 

That  is,  w^en  the  tensions  are  equal  throughout,  the  resultant  of  the 
forces  at  any  point  bisects  the  angle  made  by  the  adjacent  branches  of  the 
cord. 

219.  Let  circumferences  be  passed  through  each  vertex  and 
the  two  adjacent  ones,  and  denote  their  radii  by  r^  and  r,,  and 
let  s  be  the  length  of  one  branch  of  the  cord.     Then 

^  cos  i(//J  =  \s\  \  ,       X 

^.  cos  KVa)  =  4^.  ^      •     •     •     •     •     ^^^^^^ 

We  have  also,  from  the  figure, 

/,  cos  i(/.0  =  i^.: ) ,    .. 

h  cos  4(^3)  =  \R,.  i.  ^'^  ^ 

From  the  latter  equations  we  have 

cos  i(//.)  "  cos  \{t,t,y (^^^^ 

which  by  Eqs.  (795)  reduce  to 

r^R^  =  r^R, (798) 

That  is,  the  intensities  of  the  resultants  are  inversely  as  the  radii  of  the 
circumferences  passing  through  their  points  of  application  and  the  two 
adjacent  vertices. 

220.  From  Arts.  218  and  219  we  conclude  that  when  the  fu- 
nicular curve  has  a  constant  tension  throughout,  the  resultants  of 
the  forces  acting  at  the  different  points  are  normal  to  the  curve,  and 
their  intensities  vary  inversely  as  the  radii  of  curvature  at  their  points  of 
application. 


THE   CATENARY  CURVE.  279 


The  Catenary  Curve. 

221.  The  curve  assumed  by  a  heavy  flexible  and  inextensible 
cord  under  the  action  of  its  own  weight  is  called  a  catenary  curve. 
Assume  the  curve  in  the  plane  xz^  and  we  have  for  its  differential 
equations 

dx 
T-^^  =T^QOse^=c; (799) 


T^  =  CgdcDds  +  7;  cos  6^^  ;    .     .     .     .     (800] 


in  which  (J,  go  and  ds  are  the  density,  cross-section  and  length  of 
an  elementary  portion  of  the  curve. 

From  Eq.  (799)  we  see  that  the  horizontal  component  0/  the  ten- 
sion is  constant :  and  since  at  the  lowest  point  -^-  =  i,  the  horizon- 

as 

tal  component  of  the  tension  at  any  point  is  equal  to  the  tension  at  the 
lowest  point. 

Taking  the  origin  at  the  lowest  point,  we  have  T^  cos  6^  =  o, 
and  from  Eq.  (800)  we  have 


ds       ^e/ 


Soods (801) 


That  is,  the  vertical  co??tponent  of  the  tension  at  any  point  is  equal 
to  the  weight  of  that  portion  of  the  cord  between  this  point  and  the  lowest 
point. 

Having  the  vertical  and  horizontal  components,  the  tension 
is  readily  constructed. 

222.  The  Common  Catenary  (Fig.  83). — When  d  and  oo  are  con- 
stant the  curve  is  called  the  common  catenary^  and  we  have 

dz 

r-^=gS<as; (802) 


28o 


THEORY  OF  MACHINES. 


or,  making  ^^G?  =  i,  which  introduces  the  condition  that  the  unit 
of  length  of  the  cord  gives  a  unit  of  weight,  we  have 


as 


But 


r=:iV+7 


and  we  have 


dz  = 


sds- 


Fig.  83.  V^T^' 

Integrating,  making  s  =  o  when  z  =  o,  we  have 


^c=Vs'  -i-  c\ 


from  which  we  get 


From  Eq.  (799)  we  have 


S"  -Z^  -\-  2CZ. 


__  cds  _        cds 


T        y/-|-^^' 


and  integrating,  as  before. 


^  _[-  y  I  -I-  -  J 


~C  ' 


~c\o% 
which  is  the  equation  of  the  common  catenary. 


(803) 
(804) 

(805) 

(806) 
(807) 

(808) 


(809) 


THE   CATENARY  CURVE.  28 1 

223.  Substituting  the  value  of  s^  from  Eq.  (807)  in  Eq.  (804), 
we  have 


r=  iV-f^r7+?  =  2r  +  f (810) 

The  tension  at  any  point  is  therefore  given  by  the  ordinate  of 
the  curve  estimated  from  a  right  line  parallel  to  the  axis  of  x 
and  at  a  distance  below  the  origin  equal  to  c.  This  line  is  called 
the  directrix  of  the  curve;  it  is  readily  constructed  either  from 
the  tension  at  any  point  or  from  the  constant  horizontal  com- 
ponent of  the  tension. 

When  the  cord  is  vertical  the  directrix  passes  through  the 
lowest  point  and  is  perpendicular  to  the  cord,  and  when  the  cord 
is  horizontal  the  directrix  is  at  an  infinite  distance  below  the 
cord  and  parallel  to  it. 


APPENDIX. 


TABLE   I. 

DENSITY  AND    SPECIFIC    GRAVITY, 

I  cubic  foot  of  distilled  water  at  39.2°  F.  weighs  (:i'i.^i^  lbs. 
I     "        "  ••  "  62°  F.  •'       62.355  lbs. 


SOLIDS. 


Substances. 


Metals* — 

Aluminium 

Antimony 

Brass,  cast 

"       rolled 

"       wire 

Bronze,  gun-metal 

Copper,  cast 

'•         wrought 

Gold 

Iron,  cast 

"        "   gun-metal 

'  *      wrought 

Lead 

Platinum 

Silver 

Steel  

"     gun-metal 

Tin 

Zinc 

WOODf — 

The  density  of  a  single  variety  varies, 
but  will  seldom  differ  either  way  from  the 
tabular  values  by  more  than  ^.  Those 
given  are  average  values  for  dry,  well- 
seasoned  woods.  Green  wood  weighs  \ 
to  \  more,  and  ordinary  building  timber, 
tolerably  seasoned,  about  \  more. 

Ash 


Density. 
Water  at  62°  F. 


2.55- 
6.66 
7.8- 
8.4  ■ 

8.45- 
8.6  • 
8.8  • 

19-3  ■ 

6.9- 

7.25- 

7.6- 

"•3  ■ 

19-5  ■ 

7.8- 
7.84- 
7.2  • 
6.8  ■ 


-  2.65 

-  6.74 

-  8.4 
-8.5 

8.54 

-  8.85 

-  8.8 

-  9.0 
-19.6 

-  7.4 

-  7.4 

-  7.9 
-11.47 
-22.0 

10.5 

-  7-9 

-  7.88 

-  7.5 

-  7-2 


.6  - 


Weight  of 
Cubic  Foot 
in  Pounds. 


159-  165 
415-  420 
486-  524 
524-  530 
533 
527-  552 
536-  549 
549-  561 

1203-1222 
430-  461 
452-  461 
474-  493 
705-  715 

1216-1372 

655 
486-  493 
489-  491 
449-  469 
424-  449 


37-    44 


*  Mostly  from  Trautwine's  "  Engineer's  Pocket- Book." 

t  Mostly  from  "  Ordnance  Manual  "  and  Trautwine's  "  Engineer's  Pocket-Book. 


284 


APPENDIX. 


TABLE    \.— Continued. 


SOLID  S —  Con  tin  ued. 


Substances. 


Wood — Continued. 

Beech 

Chestnut 

Cypress 

Ebony 

Elm 

Hickory 

Hemlock 

Lignum-vitae 

Mahogany,  Honduras 

"  Spanish 

Maple  

Oak,  white 

"      live 

"      other  varieties 

Pine,  pitch 

' '       yellow  ....   

"       white 

Poplar 

Spruce 

Walnut,  black 

Miscellaneous* — 

Asphaltum 

Basalt 

Brick 

Charcoal,  soft  to  hard  woods 

' '  for  gunpowder 

Clay,  dry 

Coal,  anthracite 

"         broken 

"      bituminous 

"  **  broken 

"       lignite 

Earth,  common,  moderately  rammed 

"       mean  of  the  globe,  about 

Glass,  Aown,  average 

"       flint,  average 

*'       green,  average 

"       plate,  average 


Density. 
Water  at  62°  F. 


•5 


.6 


.6 
.7 

.65 
•7 


i.o 
2.8 
1-5 
.25 


1.3 


I.I 


-i.b 
-3. 
-2.5 
-  .6 
•38 
1.9 
-1.8 


■1.5 


■1.25 
1.5 
5.66 

2.5 

3- 

2.7 

2.7 


Weight  of 
Cubic  Foot 
in  Pounds. 


44 

31-  37 

34 

75 

37-  44 

50-  56 

28 

81 

34 

53 

37-  44 

44-  50 

62 

41-  50 

44-  50 

37 

25 

28 

28 

37 


62-112 

175-187 

94-156 

16-  37 

24 

118 

81-112 

52-  60 

75-  94 

47-  56 

69-  78 

94 


156 

187 
168 
168 


*  Mainly  from  Trautwine's  *'  Engineer's  Pocket-Book.' 


APPENDIX. 


285 


TABLE    I . — Continued. 


%0\AT>%— Continued. 


Substances. 


Density. 
Water  at  62°  F. 


Weight  of 
Cubic  Foot 
in  Pounds. 


Miscellaneous — Continued. 

Gneiss 

Graniie 

Gunpowder,    press-cake 

ordinary  grained 

Ice 

I  ndia-rubber 

Ivory 

Limestone,  marble 

common  building. 

Nitre,  crystallized 

Quartz 

Sand,  dry  to  wet 

Sandstone,  building 

Slate 

Sulphur 

Wax,  various  kinds 


2.6  -2.8 

162-175 

2.5   -2.9 

156-1S1 

1.70  -1.85 

106-115 

.875-  .900 

55-  56 

.92 

57 

.95 

59 

1.8 

112 

2.65  -2.85 

165-178 

2.4  -2.9 

150-181 

1.9 

118 

2.65 

165 

1.5  -2. 

94-125 

2.1  -2.7 

131-168 

2.7  -2.9 

168-181 

2. 

125 

.9  -I. 

56-  62 

LIQUIDS. 


Acid,  hydrochloric,  muriatic,  sat.  sol 

"       nitric,  concentrated 

"       sulphuric,  concentrated 

Alcohol,  absolute 

* '        proof  spirit 

Ether,  sulphuric,  common 

Glycerine 

Mercury 

Nitro-glycerine 

Oil,  illuminating 

"     linseed 

"     olive 

"     ("  spirits")  of  turpentine 

Water,  distilled 

"   sea 


1.21 

1-5 
1.84 

•795 

.92 

.72 

1.27 

3.6 

1.6 
.8 

•94 
.92 

.87 

1.027 


75 

93 

114 

49 
57 
44 
79 
848 
99.8 
49.9 
58.6 
57^3 
54-2 
62.4 
64.0 


286 


APPENDIX. 


TABLE     \,— Continued. 


GASES. 

Air,  dry,  at  60°  F.  and  30  in.  Bar.,  density 

"      "      "  32°  F.    -    30  "      " 


813. 


770.89 


=  .001229 


.001279 


Times 
maximum 

density 
of  water. 


Air,  dry 

Carbonic  acid,  CO2. 
oxide,  CO 

Coal  gas 

Hydrogen , 

Marsh  gas,  CH4. . . 

Nitrogen 

defiant  gas,  CqH4. 

Oxygen 

Steam  (ideal) 

"       at  212°  F..., 


Density  of 
Hydrogen 


14.422 

22 

14 

4.76-5.77 

I 

8 

14 
14 
16 

9 


Density  of 
Air 


33- 


.525 
.970 
.40 
.069 

•555 
.970 
.970 
.109 
.624 


Weight  of 
Cubic  Foot 
60°  F.,  30". 
Ozs.  avoir. 


1.226 

1.870 

1. 190 

.40-. 49 

.085 

.680 

1. 190 

1. 190 

1.360 

.765 

.592 


Weight  of 

Cubic  Meter 

0°  C,  76  cm. 

Grams. 


1293 

1973 

1255 

427-517 

89. 

717 

1255 

1255 

^435 

807 

624 


APPENDIX. 


287 


TABLE   11. 

THE  METRIC  SYSTEM. 

The  metric  system  of  weights  and  measures  is  founded  on  the  meter  as  a 
unit  of  length.     The  units  of  the  system  are  as  follows: 

Length  :  The  Meter  =  length  of  standard  bar  preserved  at  Paris. 
Area:  The  Are  =  lOO  square  meters. 
Volume :  The  Stere  =  i  cubic  meter. 
Capacity:  The  Liter  =  i  cubic  decimeter. 

Mass  and  Weight:  The  Gram  =  the  mass  or  weight  of  i  cubic  centimeter  of 
distilled  water  at  the  temperature  of  maximum  density. 

It  is  a  decimal  system.  The  prefixes  denoting  multiples  are  derived  from 
the  Greek,  and  are:  deka,  ten;  hecto,  hundred;  kilo,  thousand;  and  myria,  ten 
thousand.  Those  denoting  sub-multiples  are  taken  from  the  Latin,  and  are: 
deci,  tenth;  centi,  hundredth;  and  7nilli,  thousandth. 

The  following  table  includes  all  the  measures  of  the  system  in  use: 


No.  of 

the 
Unit. 

Length. 

Area. 

Volume. 

Capacity. 

Mass  and 
Weight. 

lOOOO 

Myriameter. 

Myriagram. 

1000 

Kilometer. 

Kilogram,  kg. 

100 

Hectometer, 

Hectare,  ha. 

Hectoliter,  hi. 

Hectogram. 

10 

Dekameter. 

Dekastere. 

Dekaliter,  dal. 

Dekagram. 

I 

Meter,  m. 

Are,  a. 

Stere,  s. 

Liter,  1. 

Gram,  g. 

.1 

Decimeter,  dm. 

Declare. 

Decistere. 

Deciliter,  dl. 

Decigram,  dg. 

.01 

Centimeter,  cm. 

Centiare. 

Centiliter,  cl. 

Centigram,  eg. 

.001 

Millimeter,  mm. 

Milligram,  mg. 

The  are  and  its  derivatives  are  used  only  for  land  measure.  In  other  cases 
area  is  expressed  in  terms  of  the  square  whose  side  is  a  measure  of  length — 
e.g.,  square  meter,  m';  square  centimeter,  cm'-^,  etc. 

The  stere  is  rarely  used  except  in  measuring  firewood.  In  other  cases  a 
cube  whose  edge  is  a  unit  of  length  is  used — e.g.,  cubic  meter,  m^;  cubic 
dekameter,  dm^  etc.     Cubic  dekameter,  cubic  hectometer,  etc.,  are  not  used. 

A  Mikron,  //,  =  .001  mm. 

A  Tonne,  t,  or  millier,  =  1000  kg. 

A  Metric  Quintal,  q,  =  100  kg. 


288 


APPENDIX. 


to 

It 


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APPENDIX. 


289 


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290 


APPENDIX. 


TABLE   IV. 

GRAVITY, 

g  =  Acceleration  due  to  gravity,  in  feet  per  second. 

L  =  Length  of  simple  seconds  pendulum,  in  feet. 

A  =  Latitude. 

k  =  Height  above  sea-level,  in  feet. 

g  —  32.173  —  0.0821  cos  2X  —  0.000003/^.* 

L  =  3.2597  —  0.0083  cos  2/1  —  0.0000003>^.* 

Values  of  ^  =  32.173  —  0.0821  cos  2A  and  L  =  3.2597  —  0.0083  cos  2A. 


Latitude. 

£r. 

L. 

0' 

32.091  f.  s. 

3-2514  f. 

5 

32.092 

3-2515 

ID 

32.096 

3-2519 

15 

32.102 

3-252S 

20 

32.110 

3-2533 

25 

32.120 

3-2544 

30 

32.132 

3-2556 

35 

32.145 

3.2569 

40 

32.159 

3.2583 

45 

32.173 

3.2597 

50 

32.187 

3.2611 

55 

32.201 

3.2625 

60 

32.214 

3.2638 

65 

32.226 

3.2650 

70 

32.236 

3.2661 

75 

32.244 

3 . 2669 

80 

32.250 

3.2675 

85 

32.254 

3.2679 

90 

32.255 

3.2680 

The  value  of  g  is  affected  to  some  extent  by  the  character  and  arrangement 
of  the  local  geological  strata.  The  variation  from  the  tabular  value  may  be  as 
great  as  10  units  of  the  last  place  of  figures,  but  rarely  exceeds  5  of  these  units. 


*  Encyclopaedia  Britannica,  art.  Gravitation. 


APPENDIX. 


291 


TABLE  V. 

FRICTION  * 


Substances. 

Angle  of 

Repose. 

0 

Coefficient  of 
Friction. 

-f- 

1 

Vi  +/« 

14     -26.5 
II-5 
II. 5-31. 
26.5-31. 

11-5 

II. 5-14. 

26.5 

16.5 

8.5-11.5 

16.5 

4.-4.5 

3. 

1.7 

.25-.5 
.2 

.2  -.6 

.5  -.6 

.2 

.2  -.25 

•5 

.3 
.I5-.2 

.3 
.07-. 08 

.05 

.03 

.24-. 45 

.20     1 
.20-. 51 
.45--5I 

.20 
.20-. 24 

.45 

.29 
.15-. 20 

.29 
.07-. 08     j 

.05 

.03 

i 

"      "       "        soaoed 

Wood  on  metals,  dry 

Metals  on  oak,  dry 

' *       * '     * '      soaped  

Hempen  cord  on  oak  dry    

"      "     "      wet 

Metals  on  metals,  dry 

"       "         "       wet     

Smooth  surfaces,  occasion'y  greased 
'*              **          continually      " 

best  results 

These  values  are  for  low  velocities  and  pressures  at  ordinary  temperatures. 
The  coefficient  for  smooth  metal  bearings,  well  oiled,  varies  somewhat  with  the 
pressure  and  velocity,  being  generally  less  than  the  above.  It  also  varies  con- 
siderably with  the  temperature,  which  affects  the  lubricant.  In  favorable  cases 
it  has  been  as  low  as  .002  [Thurston]. 


Mostly  from  Rankine's  "  Rules  and  Tables,"  and  Trautwine's  "  Engineer's  Pocket-Book.' 


592 


APPENDIX, 


TABLE  VI. 

STIFFNESS  OF  CORDAGE  FOR    WHITE  AND    TARRED 

ROPE. 

Morin's  Formulas.* 

6"  =  — — =  ^(0.002148  4-  0.001772W  -["  0.0026256  W)  for  white  rope; 

K  =  «(o.oo2i48  -f-  o.ooi772«),  and  /  =  «(o.oo26256). 

S  =  — ^ =  —(0.01054  4-  o.oo25«  -\-  0.003024  ^)  for  tarred  rope; 

K  =  «(o.oio54  -f-  o.oo25»),  and  /  =  «(o.oo3024). 


Values 

OF  ^AND  /IN  Lbs., 

FOR  ROPK  WOUND  ON  A 

XLE  I  Foot 

IN  Diameter. 

j 
i 

No.  of 
1  Yarns. 

Ordinary  White  Rope. 

Tarred  Rope. 

Circum- 
ference 
in  inches 

Natural 
Stiffness. 

Stiffness  due 

to  Tension  of 

lib. 

/. 

Circum- 
ference 
in  inches 

Natural 
Stiffness, 

Stiffness  due 

to  Tension  of 

lib. 

/. 

i 

= 

/if  in  lbs. 

= 

A' in  lbs. 

.4524  v^ 

.5378  Vn. 

6 

1. 10 

0.07668 

0.015754 

1.32 

0   15324 

O.O18146 

i       9 

1.36 

0.16286 

0.023630 

1. 61 

0.29736 

0.027219 

12 

1.57 

0.28094 

0.031507 

1.86 

0.48648 

0.036292 

15 

1.74 

0.43092 

0.039384 

2.08 

0.72060 

0.045365 

18 

1.92 

0.61279 

0.047261 

2.28 

0.99972 

0.054438 

21 

2.08 

0.82656 

0.055138 

2.46 

1.32384 

O.063511 

24 

2.21 

1.07222 

0.063014 

2.64 

I .69296 

0.072584 

27 

2.35 

1.34978 

0.070891 

2.80 

2. 10708 

0.081657 

30 

2.47 

1.65924 

0.078680 

2  95 

2 . 56620 

0.090730 

33 

2.60 

2.00059 

0.086645 

3.09 

3.07032 

0.099803 

36 

2.72 

2.37204 

0.094522 

3.23 

3.61944 

0.108876 

39 

2.84 

2.77888 

0.102398 

3.36 

4.21356 

0.117949 

42 

2.94 

3.21602 

O.IIO275 

3.48 

4.85268 

0.127022 

45 

3.05 

3 • 68496 

O.I18152 

3.61 

5-53680 

0.136095 

48 

3-17 

4.18579 

0.126929 

3.73 

6.26592 

O.145168 

51 

3-26 

4.71852 

0.133906 

3.84 

7.04004 

o.T542+r 

54 

3-35 

5.28314 

O.141782 

3-95 

7.85916 

0.163314 

57 

3  45 

5.87966 

0.149659 

4.07 

8.72328 

0.172387 

60 

3.54 

6 . 50808 

0.157536 

4.17 

9.63240 

0.181460 

*  Adapted  from  Morin's  formulas,  "  Cours  de  M^canique,"  vol.  ii.,  Dulos,  pp.  193,  194. 


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